tl;dr

Since when? –– Since the middle ages. // Since when approaching modern concepts? –– Since the late 19th century. // Since when with for historians directly usable data on the stated subject matter? –– Since the 1930s.

Necessary prolegomena:

Our view on the subject is clouded by most modern conceptions about pubic opinion and its measurement. How do we apply these concepts of ours into former times? It is difficult enough to get a coherent statement and opinion out of a contemporary subject when asking him about anything. Try to reduce this babble that population masses represent to something comparable and the precision and intricacies get quite blunt. A method to gauge public opinion or the opinion of the public can be seen in Roman times, when people debated on fora and displayed approval in games and theatre –– not just about contestants or thespians, but –– of people, politics and politicians that were present or absent.

The origins of our modern conception of public opinion are usually traced to liberal democratic theories of the eighteenth century, with precursors reaching all the way back to ancient Greece (Palmer, 1936). And yet the connections between empirical public opinion research and political theory have been remarkably loose.

The concept of public opinion emerged during the Enlightenment, but the separate concepts of the public and opinion have much older histories, each with a range of meanings that continue to inform their use to the present day (Price, 1992).

The compound concept public opinion came into widespread use only in the eighteenth century and as the product of several significant historical trends, primarily the growth of literacy, expansion of the merchant classes, the Protestant Reformation, and the circulation of literature enabled by the printing press. An ascendant class of literate and well-read European merchants, congregating in new popular institutions such as salons and coffee houses and emboldened by new liberal philosophies arguing for basic individual freedoms, began to articulate a critique of royal absolutism and to assert their interests in political affairs (Habermas, 1962/1989).

For centuries, elite opinions have been gauged by messengers or spies, or by searching letters, diaries, or pamphlets. Opinions of illiterate masses, to the extent they were not totally ignored, had been gauged by thumbs ups or thumbs downs in local stadiums and rinks. Countrywide public opinion polling of the general population by statistical methods has a shorter history. It is a child of the American newspaper world, born in the 1930s. In this chapter, we meet the launching actors, their ideas about the nature and use of opinion reporting, and their methods of researching opinions.
_{Vincent Price: "The Public and Public Opinion in Political Theories", Donsbach/Traugott(Eds): "The SAGE Handbook of Public Opinion Research" 2008.}

What this entails is that just "asking around" is not considered a valid way of measurement at all. Thus the following history of the field delimits the search for 'real public opinion polls':

The empirical tradition in opinion and attitude research began modestly enough in Germany with simple laboratory experiments on problem solving, in which the notion of "mental sets" was carved out. It gained strength from the work of the Chicago school of sociologists, which brought the study of attitudes and values into play. Immediately thereafter, the psychometricians under the leadership of Thurstone introduced the portentous problem of measurement. And finally came the public opinion research people who, on the one hand, narrowed the conceptual range but, on the other, greatly extended the field of practical applications.
_{Paul Lazarsfeld: "Public Opinion and the Classical Tradition", Public Opinion Quarterly, 1957.}

Nineteenth-century surveys tended to resemble censuses in being attempts to cover the total populations of communities. For example, Booth s late- nineteenth-century survey of the poor of London was a block-by-block, house- hold-by-household survey, more or less systematic in its coverage of topics (Fried & Elman, 1968). Dubois s 1892 survey (1967) of the Philadelphia Negro community was a household-by-household canvass using schedules that resem- bled closely the forms used by the 1890 census covering basic demographic and employment data.
The social surveys that were at the heart of the early twentieth-century "survey movement" were also total censuses of the cities studied, merging census data with special surveys of topics such as housing conditions that were not covered in the decennial censuses. The work of the early human ecologists of the Chicago School was also based on the same model: Thrasher (1927) attempted to map the location of all the juvenile gangs in Chicago; Reckless (1933) surveyed all the houses of ill fame for his spotmaps showing the locations of bordellos in Chicago (and thereby spawning a plethora of satirical aphorisms on why sociologists need foundation grants to find things any other citizen would know "naturally").
Similarly, preelection surveys conducted by newspapers to forecast election results were based on the notion that the bigger the sample, the better the forecast. The Literary Digest straw polls of telephone subscribers were based on returns of millions of ballots mailed to all telephone subscribers in the United States. Newspapers sent reporters and hired canvassers, with paper ballots and ballot boxes, to busy intersections to intercept voters to conduct straw votes. Those who conducted the straw votes based their claim for fore- casting accuracy on the numbers of persons polled. Note that the rationale for forecasting accuracy was not a sampling one, it was based on the notion that the larger the N, the greater the accuracy, a notion that is only true if N is quite close to the universe size.
The consumer marketing research that started after World War I initially operated on a different model that was not imitative of censuses. Rather, the model was that of the psychophysical laboratory in which a small number of subjects are brought to a location for standardized testing. In the laboratory model, the processes being investigated were pan-human and hence any assemblage of subjects was as valid as any other assemblage. Early product testing asked assemblages of consumers to express their comparative preferences for an array of packages or asked consumers to try out a new brand. Consumers were located through stores or social clubs, or they were simply asked to volunteer through ads in newspapers. Modern surveys evolved out of these roots.
_{Peter H. Rossi James D. Wright Andy B. Anderson: "Handbook of Survey Research", Academic Press: London, New York, 1983.}

The concept of "the public opinion" was in this view simply never measured before. What we find before is methodological precursors, at best. Or something called a "poll" or "survey" that just represents a very small fraction or subset of "the public", and even there the scientific goal of "the truth about reality" is far from reached and any form of representativeness a mere statement without grounds, unattainable with these methods without a total census. That is the problem with straw polls: they are more of a fun game than a poll of public opinion. These straw polls could produce back then and can produce now only something that people at the time believed to be true. That can be seen as either severely limiting or interesting in itself, but it is quite important to draw this distinction.

While before much more indirect measures were taken, inferences from newspaper writers or editors, data collected from spies, messengers recorded that "talk of the tavern" and so on; early forms of "asking around" are to be found, without any still standing claim to scientific validity of course. But they might serve as illustrations, with a host of caveats for us to equate them as 'expressions of public opinion'. They are expressions of that, if we keep in mind that they are distorted subpopulations, with sample errors and all sorts of other biases. Even the Harrisburg example is not the public, as it was distorted by self-selection, literacy and newspaper subscription, to name just the biggest obstacles that hinder an approach to call the results anything "about the public". That kind of inquiry about what we now would define as "the public" simply did not feature before the First World War.

Therefore, one of very many examples of these inadequate precursors is from German speaking lands:

Isidor Singer was a journalist that was engaged in dealing with the "Jewish question". That is a simple and yet contentious question that was debated publicly. He actually did ask around: "What do you think of the Jews?" This was asked in this form in 1885 and 54 intellectuals responded. (Cf: Thomas Gräfe: "Was halten Sie von den Juden?": Umfragen über Judentum und Antisemitismus 1885-1932", BoD: Norderstedt, 2018.)

Another would be

Arthur Kirchhoff: "Die akademische Frau: Gutachten hervorragender Universitätsprofessoren, Frauenlehrer und Schriftsteller über die Befähigung der Frau zum wissenschaftlichen Studium und Berufe", Hugo Steinitz Verlag: Berlin, 1897 (PDF). (The Academic Woman: Expert Opinions of Outstanding University Professors, Women's Teachers and Writers on the Qualification of Women for Academic Studies and Professions".)

Where all professors were asked and in that book the preface would represent a summary of findings from this kind of "public opinion poll".

Both examples represent a form of public opinion that is qualitative and only roughly quantified into "majority or not", no pseudo exactness in numbers.

Another very nice example, and quite an early one, is a questionnaire regarding standardised questions apparently on the opinions of clerical matters, but not exclusively.
It was sent throughout the realm by Charlemagne in the year 811.

With a list whose character resembles that of a discussion guide for an intensive interview, the emperor wanted to instruct the spiritual and secular dignitaries of the empire assembled at the Reichstag, but it is also apparent that at least some of the questions were asked out of interest in information about the mood of the population in the empire. These questions were probably passed on to the princes and clergy in the provinces. Another document from the year 811 indicates that actual answers to similar question lists were received at the imperial court. Both texts document a procedure for obtaining information that could be described as one of the first attempts at an opinion poll.
_{Thomas Petersen et al.: "Der Fragebogen Karls des Grossen. Ein Dokument aus der Vorgeschichte der Umfrageforschung", KZfSS Kölner Zeitschrift für Soziologie und Sozialpsychologie, December 2004, Volume 56, Issue 4, pp 736–745.}

And despite his own Alcuin waring Charles earlier that Nec audiendi qui solent dicere, Vox populi, vox Dei, quum tumultuositas vulgi semper insaniae proxima sit. An adage that pertains equally today or a hundred years ago when "asking around" or vox pop is more constructing public opinion or justifying what a newspaper writes than a genuine display of what the population thinks.

An even better summary of public opinion polls with regard to the question is presented in:

There are many good polls. Their conclusions can be believed. There are, however, also many bad polls. It is better to ignore them. Unfortunately, it is not always easy to separate the wheat from the chaff.

Opinion polls can be traced back to ancient Greece. Around 500 BC, many city–states had a form of government based on democratic principles. All free native adult males could express their opinion on topics such as declaring war, dispatching diplomatic missions, and ratifying treaties.

There are polls, and there are surveys. There are no essential differences between polls and surveys. Those are both instruments to collect data from a sample by means of asking questions. There are, however, some practical differences. A poll is often small and quick. There are only a few questions (sometimes even only one) that have to be answered by a small sample of, say, 1000 people. A poll contains mainly opinion questions. A survey can have both opinion and factual questions.

For a long period in history, data collection was based on complete enumeration of the target population. Every person in the target population had to provide information. The important idea of sampling emerged only at the end of the nineteenth century. It had taken many years before this idea was accepted.

You can see opinion polls as surveys that measure attitudes or opinions of a group of people on political, economic, or social topics. The history of opinion polls in the United States goes back to 1824. In this year, two newspapers, the Harrisburg Pennsylvanian and the Raleigh Star, attempted to determine political preferences of voters prior to the presidential election of that year. These early polls did not pay much attention to sampling aspects. Therefore, it was difficult to establish the accuracy of their results. Such opinion polls were often called straw polls. This expression goes back to rural United States. Farmers would throw a handful of straws into the air to see which way the wind was blowing. In the 1820s, newspapers began doing straw polls in the streets to see how political winds blew.
It took until the 1920s before more attention was paid to sampling aspects. At that time, Archibald Crossley developed new techniques for measuring American public’s radio listening habits. Moreover, George Gallup worked out new ways to assess reader interest in newspaper articles.
_{Jelke Bethlehem: "Understanding Public Opinion Polls", CRC Press: Boca Raton, 2018. From the first two chapters, p 1–31.}

The importance being seen yet the means not developed for centuries is seen in:

In 1888, the perceptive British journalist and author James Bryce would claim that “in no country is public opinion so powerful as in the United States” (1900). He also noted, “the obvious weakness of government by opinion is the difficulty of ascertaining it.”
Of those writing before the development of the modern opinion poll, perhaps the most influential critic of public opinion was Walter Lippmann (1922, 1925). Like many of the founders, Lippmann believed mass opinion was subject to passions that could be induced by elite propaganda. He was convinced that the manipulation of public opinion by those opposed to the League of Nations was responsible for the tragedy of America’s failure to join after World War I. Famously, Lippmann perceptively observed that the images of politics received by the public are not direct pictures of events, immediate experiences of action, or provable economic and social theories. Rather, they are “pictures in people’s heads” generated by political interests to benefit their cause. In a prescient analysis of major findings by modern survey research, Lippmann challenged traditional democratic theory and its notion of an informed and rational public basing opinions on a considered judgment of the facts. He argued that the average person had little time for affairs of state and would rather read the comics than consider the pros and cons of weighty political issues. It should not be expected, therefore, that the mass public be competent in matters of state. Lippmann’s prescription for democracy was for the public to choose leaders but for public policy to be developed and implemented by scientifically oriented experts.

Sociologist Herbert Blumer and political scientist Lindsay Rogers soon launched frontal assaults on the opinion poll and its implications. Blumer (1948) asserted the “one person, one vote” definition of public opinion inherent in polls was precisely what public opinion was not. Public opinion could not be reduced to a nose count of citizens. Rather, it was the interactions and communications among functional groups that percolated through society and came to the attention of government. These interactions and communications were not aggregations of individual opinions but “an organic whole of interacting, interrelated parts.”
_{Robert S. Erikson & Kent L. Tedin: "American Public Opinion. Its Origins, Content and Impact", Routledge: London, New York, ⁹2015.}

This topic is difficult to research and as late as the 1960s still considered "new". If the research subject is "British public opinion" then at Princeton we find:

Public opinion polls for historians Public opinion (London, England) [microform]. London : G. Cole, 1861-1951. RECAP MICROFILM S01253

With a quite severely outdated methodology (and difficulties in gaining access). And later

World political opinion and social surveys. Series one: British opinion polls. [microform]. Reading, Berkshire : Research Publications, 1990- MICROFICHE 2282

British opinion polls, 1960-1988: a comprehensive subject and names index of the complete range of surveys and opinion polls by Gallup, NOP, MORI, and Harris in Great Britain, 1960-1988 Edited and compiled by David Tyler. Reading, Berkshir : Research Publications, 1990. (Film B) HN400.P8 B757 1990

British public opinion: a guide to the history and methodology of political opinion polling. Robert M. Worcester. Oxford; Cambridge, MA: B. Blackwell, 1991. Firestone Library (F) HN400.P8 W67 1991 A concise history of political opinion polls in Britain from 1937-1987.

British comment on the United States: a chronological bibliography, 1832-1899. Ada B. Nisbet ; edited by Elliot J. Kanter ; with a foreword by Asa Briggs. Berkeley : University of California Press, c2001. Annex A, Forrestal (TEMP): Z6465.U5 N58 2001

Given these problems, the National Archives of the UK offer an educational resource to tackle the problem:

Did public opinion drag the UK into the war (1914)?

With the following sources to analyse:

\1. Resolutions of Labour Party branches, 1914 –– 2. Diary of a factory worker, September 1914 –– 3. Photographs of volunteers, 1915 –– 4. Extract from 'War Against War', 1914 –– 5. Posters on recruitment, September 1914 –– 6. Note on recruitment, December 1914

Again pertaining to the World War, the general picture of public opinion in Germany is painted as elated inebriation in August that displayed an inherent German eagerness and enthusiasm for going to war, finally. We saw it on public places and in the papers. Yet, the biggest political party in Germany practically silenced its own constituency on the matter by voting in support of the war. Historians find largely an attitude of worry and even depression as the dominating opinion among those.
That leads again to the question of what public opinion is: publicly available opinion, published opinion or the opinions of the major factions of a society.

_{Wolfgang Kruse: "Krieg und nationale Integration. Eine Neuinterpretation des sozialdemokratischen Burgfriedensschlusses 1914/15", Klartext Verlag: Essen, 1993. Christian Geinitz: "Kriegsfurcht und Kampfbereitschaft. Das Augusterlebnis in Freiburg. Eine Studie zum Kriegsbeginn 1914", Klartext: Essen, 1998. Gerhard Hirschfeld & Gerd Krumeich& Irina Renz (eds.): "Enzyklopädie Erster Weltkrieg", Schöningh: Paderborn, 2014.}

As no polls or even a referendum was conducted and elite expressions or some public acclamations were used to declare that to be an expression of public opinion and the general will, if not fate and inevitability, it can be conversely said that any Nixon-like argument about a silent majority and their opinion would have indeed ended the war before Christmas.