Ancient Greece was always divided into self-ruled city-state kingdoms. People of each of these had their own distinct culture and they traced descendance with a particular god or any other figures like a hero. Accordingly they believed that the deity of their city protects and helps develop their city. So they mainly worshipped the gods of their own city.

However just as a Christian would visit any ancient church in another city, belonging to any other sect of Christianity, similarly people worshipped the god of a place when they visited that place, the way tourists of all religions would visit the Basílica de la Sagrada Família in Spain.

Another important thing is that people belonging to particular professions ( often a family profession for successive generations) would worship the god of their particular profession. Sailors worshipped Poseidon, whereas students, scientists and philosophers worshipped Athena. If you went to their place they would suggest you to pray or give offerings to their god before doing their particular work. Like they could ask you to pray once to Aries before going to war.

The bottom line: There couldn’t have been forcing one to pray or give offerings to certain gods only. However u could be suggested to pray one over the other. Often it was also based on personal liking of a god.

The Spartans were, in fact, noted for their religiosity in ancient Greece. However, there is no sign that Demeter or Dionysus had any cult in Sparta. This reflects that you didn't, generally, worship any god that had no relevance to your life.

As a general rule, you worshipped the gods after the manner of your fathers, and if your neighbor's father had worshipped differently, he worshipped that way, and apparently the gods were happy with it, since they didn't destroy you.

The chief objection to other gods was not that they were false but that they were foreign. Every now and again the worship of Isis was forced out of Rome as a foreign god; as an Egyptian goddess, her worship was suited for Egyptians, not Romans.

It was very confusing by our standards. Paganism in the Roman Empire by Ramsay MacMullen is good for more detail

There is a major definition problem here.

The cultures you reference did not draw a distinction between church and state. To refuse to offer incense to the patron god of the state was to refuse the legitimacy of the state. One metaphor might be that to refuse incense was like refusing to pay taxes - god and the state deserved their due.

In general, these polytheistic cultures didn't care who you worshipped, but they did care if you openly and publicly denied that the patron god existed - that was an act of civil disobedience that no society could ignore.

The intolerance wasn't the polytheists, it was the primitive Christians, who refused to acknowledge that their neighbor's worship & participation in the state was legitimate. Primitive Christians denied public ritual, and refused to participate in communal government driven activities. That intolerance escalated to civil punishment.

SideNote: it is inaccurate to call state sponsored religion "pagan" - pagan religion connotes rural/uncultured religious ritual.

Aside: THere is a good presentation of a related question, Did the Great Heathen Army persecute Christians; different era, different polytheistic pantheon, but the same confusion over the concept of tolerance.

Update

To answer your question If they "did not draw a distinction between church and state," that would also seem to imply that the state was a god. Is that true? the answer is no. le etat cest moi per Louis XIV may be the appropriate analogy once emperors became the norm. "The state" as we came to know it in the post enlightenment wasn't quite it. And in the case of Rome, once Julius Caesar and the Imperial age overtook the Republic for good, the linkage between the Emperor and the State became more pronounced.

That is a reasonably subtle question, and deserves an answer that is both better reasoned and better researched than I'm capable of (particularly at this moment). I think that they would generally disagree that the state was a god, but that they would agree that the state could only survive with the active cooperation & sponsorship of a god. I think that 95% of the population would agree that gods controlled the unseen forces that were responsible for the success or failure of anything beyond human control. Every state/polis/civic entity/institution was sponsored by a god. The institution flourished with the god's favor and suffered for the god's ire.

I'm going to draw a broad analogy here - this will not hold up for analysis, and in no way do I mean to disrespect, but imagine that there were an ethnic group in modern day america that refused to say the pledge of allegiance, to pay normal deference to the flag, refused to participate in judicial proceedings because they included the flag, refused to enter public buildings that displayed the flag, refused to send their children to public schools (because the school displays the flag). This group would not, under penalty of prosecution, perform even the most minimal acts of respect for the flag (standing, saluting, etc.) Imagine that they rejected the compromises of groups like the Jehovah's Witnesses who find compromise positions that endorse the state while rejecting the practices that they find idolatrous. Such a group would be subject to a great deal of scrutiny to try to understand their contempt for our shared symbol. THe early Christians rejected the civic symbols of their neighbors - they went even further and asserted that respect for national symbols was idolatrous and sinful and imperiled the eternal soul.

Like I said, this is not a precise metaphor - nobody today believes that the flag's happiness is required for national prosperity. But if such a group existed (and I am not interested in examples or counterexamples), they would be the intolerant ones, not the flag wavers.

This is veering away from history rapidly.

It is entirely possible to find ancient Greeks and Romans discussing this very matter. My amateur opinion is that it was a favorite parlor game to try to map the gods of one civilization to those of another and even discover new ones. I'll provide several examples of this from Herodotus. People are skeptical of Herodotus in terms of factual accuracy, but his work does represent the opinion of one influential Greek citizen. I'm going from this copy on the web.

Regarding the Scythians:

They worship only the following gods, namely, Vesta, whom they reverence beyond all the rest, Jupiter, and Tellus, whom they consider to be the wife of Jupiter; and after these Apollo, Celestial Venus, Hercules, and Mars. These gods are worshipped by the whole nation: the Royal Scythians offer sacrifice likewise to Neptune. In the Scythic tongue Vesta is called Tabiti, Jupiter (very properly, in my judgment) Papaeus, Tellus Apia, Apollo Oetosyrus, Celestial Venus Artimpasa, and Neptune Thamimasadas. They use no images, altars, or temples, except in the worship of Mars; but in his worship they do use them.

The Persians:

The customs which I know the Persians to observe are the following: they have no images of the gods, no temples nor altars, and consider the use of them a sign of folly. This comes, I think, from their not believing the gods to have the same nature with men, as the Greeks imagine. Their wont, however, is to ascend the summits of the loftiest mountains, and there to offer sacrifice to Jupiter, which is the name they give to the whole circuit of the firmament. They likewise offer to the sun and moon, to the earth, to fire, to water, and to the winds. These are the only gods whose worship has come down to them from ancient times. At a later period they began the worship of Urania, which they borrowed from the Arabians and Assyrians.

Assyrians:

A woman who has once taken her seat is not allowed to return home till one of the strangers throws a silver coin into her lap, and takes her with him beyond the holy ground. When he throws the coin he says these words- "The goddess Mylitta prosper thee." (Venus is called Mylitta by the Assyrians.)

The Massagetae (who were apparently nomads from the Asian steppe):

The only god they worship is the sun, and to him they offer the horse in sacrifice; under the notion of giving to the swiftest of the gods the swiftest of all mortal creatures.

And the Ethiopians:

The only gods worshipped by the inhabitants are Jupiter and Bacchus, to whom great honours are paid. There is an oracle of Jupiter in the city, which directs the warlike expeditions of the Ethiopians; when it commands they go to war, and in whatever direction it bids them march, thither straightway they carry their arms.

Herodotus, at least, seems to have a very expansive sense of the number of gods. He takes joy in the discovery of new gods and new rituals, and believes that other cultures have good ideas about religion and powerful connections to the gods. He seems to judge those who strictly limit the number of gods to be backward, missing out on the fulfillment possible in a broader religious practice. That said it seems like every city and individual had a subset of deities that they were partial to, and considering the innumerable nature deities (such as every river) it would be impossible to worship them all.

Like other answerers have said, it seems there was always room for more gods, but it may have been bad manners to suggest some god didn't exist.

Did they tolerate those who only believed in some of their gods?

I don't see how it is possible - how does one believe in, say, Mars but not in his father Jupiter? Mars is defined as a Jupiter's son! Given that the Greek/Roman pantheon is a sex, jealousy & violence -obsessed dysfunctional family, it makes no sense for a person to deny divinity of any family member, but a lot of sense to worship a specific character.

Essentially, gods were the celestial analogue of patronage: if one shows loyalty to another patron/god, one's own will be pissed more than the other pleased, but if one denigrates another patron/god, the other will be pissed more than one's own pleased. Thus the rational position is "I serve my own god but do not raise up against others".

Did the Greeks or Romans thus esteem tolerance and a plurality of gods (polytheism) more highly than the gods themselves?

I don't see how you can use "tolerance" and "Greeks or Romans" in the same sentence. Tolerance of a worshipper of Jupiter towards a worshipper of Mars is similar to mutual tolerance of Star Trek fans who admire different characters. They will happily and ruthlessly gang up on a Tolkienist. :-)

Monotheists (like Jews and Christians) deny all the "pagan" gods en mass. The Deity they worship is completely outside of the Greek/Roman narrative.

The demands you mention (offer incense to a specific god worshiped in this particular locale) is based on the idea that refusal will offend the mob watching the proceedings and make the forthcoming execution all the more justified in its eyes.