Which culture used no personal names?

This would seem to be a piece of folklore.

Anthropologists have not found a single society which does not use personal names in some form; they are a human universal. However, the forms that these names take and the ways in which they are bestowed and used vary between cultures.

Source: Abstract from Ellen S. Bramwell, 'Personal Names and Anthropology'. In Carole Hough (ed.), 'The Oxford Handbook of Names and Naming' (2016)

William Bright (1928-2006), formerly a linguist at the University of California and then at the University of Colorado, also covered this in a 2003 article What IS a Name? Reflections on Onomastics:

There is a piece of folklore current among anthropologists regarding the question of whether personal names exist in all societies. So far I have not been able to trace this to a printed source, but it is somewhat as follows: Somewhere in the world there is a society where people live in very small, isolated communities. In such a community, people have no personal names; i.e., individuals have no name which other people use to refer specifically to them. Instead, they are referred to by descriptive expressions, e.g., ‘the blacksmith’ or ‘the man who lives by the stream’. A woman will be referred to as, e.g., ‘the blacksmith’s wife’. Children will be referred to by expressions such as ‘the blacksmith’s elder daughter’; when this daughter gets married, she may be referred to as, e.g., ‘the wife of the man who lives by the stream’.

He concludes that no such society exists, that

any anthropologist who might have reported such a community was misled by the operation of taboos on uttering personal names. I suggest, in fact, that the use of personal names, having varying levels of descriptiveness, is a sociolinguistic universal of the human species.

Bright points out that

in non-literate societies, where names remain unwritten, there is greater variety in naming customs (cf. the anthropological studies in Tooker 1984). A child may be given a “real” name at birth, but this may be kept a secret throughout life. Elsewhere, such a “real” name may be publicly known, but not used for everyday purposes; most of the time, a nickname—perhaps descriptive, e.g., Shorty—may be used. A person may be called by different names at different periods of life, or by different people under changing conditions. Use of certain names under particular circumstances may be forbidden by religious taboo; or then again, such names may be replaced by descriptive nicknames. Because of these factors, it may be difficult for the outside investigator of such a society to determine what a person’s “real” name is, or even what name is commonly used in the community; taboos are likely to be especially strict when one is talking to outsiders.

An Australian Institute of Family Studies (AIFS) article Aboriginal families in Australia (1995) by Professor Colin Bourke and Eleanor Bourke illustrates how confusing the issue of names can be:

In many Aboriginal societies personal names were rarely used. People were addressed by kinship terms. Some were referred to as being someone else's son or daughter. In such societies personal names were seen as being part of that person and were used with discretion. This was often manifested by a deceased person's personal name being removed from that language for some considerable time. Most languages had a word meaning 'no name' which was used to refer to those persons who had the same name as a recently deceased person.

Also, Wulf Schieffenhövel's article Reactions to cultural change (1997) in the journal Civilisations relates the problems the author encountered with the Eipo in the Highlands of West-New Guinea:

Each of them also seemed to answer my question: «An si?», what's your name? It was only later when I had mastered the first steps into this highly structured language, that I realized what these names meant: Na-si-gumnye, Na-si-walwal, Na-si-urang, «I-the-one-who-has-no-name», «My-name-I-don't-know', «I-am-somebody-else». To give away one's dibe si, the «real name», exposes oneself to possible harm.

Other studies from other regions of the world also show the importance and sometimes confusing (to outsiders) use of names: the Ilongot in the Philippines (where people "acquire and lose" names during their lifetime), the Kadayan in Brunei and Malaysia, societies in Central Brazil, and the Mohawk people. These studies illustrate the "diversity of the processes involved in naming and using names".

Stephen Wilson, in The Means of Naming: A Social and Cultural History of Personal Naming in Western Europe (UCL Press, 1998), outlines just how important names have always been in societies, even though (as he notes elsewhere) historical documents have often not recorded them.

In all societies...and more so in complex ones, a single individual is known by a variety of names depending on the role he or she is playing and the milieu of reference. So there will be one name used by close relatives of origin, another by spouses and lovers, another by children, another by friends, another in public, another at work, and so on. Again, names here serve as a kind of social map, placing individuals in the broader multi-dimensional landscape.

Wilson's primary area of study was Western Europe, but he also cites Gregory Bateson's work, Naven (1936), on the Iatmul of New Guinea to show how important names are to both individuals and societies:

Every spell, every song…contains lists of names. The utterances of shamans are couched in terms of names… Marriages are often arranged in order to gain names. Reincarnation and succession are based upon the naming system. Land tenure is based on clan membership and clan membership is vouched for by names.

To summarize, all societies use personal names, but

Names are given to people at different stages of life; they change or remain constant; they contain different elements; they connect with relatives or tribes or they do not; they are used freely or they are kept secret.

Source: Ellen S. Bramwell, 'Personal Names and Anthropology'. In Carole Hough (ed), 'The Oxford Handbook of Names and Naming' (2016)

(All emphasis is mine)