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