There's a certain bias in that assertion that "human names aren't unique", especially in the John Smith example. A legal name is at least partly social construct meant to interact with a government bureaucracy. Compare the surname situation in Japan, which had most people come up with their own surnames a scant century and a half ago and consequently has a huge diversity of surnames, with China, which has been undergoing surname extinction for millenia. Ultimately this has more to do with large legal and kinship systems more than purely social identification.
https://en.wikipedia.org/wiki/Japanese_namehttps://en.wikipedia.org/wiki/Galton%E2%80%93Watson_process
Human names in an oral culture on the smaller family-and-friends scale seem like they'd be much more diverse, unique, and informal, with names that are more like nicknames than legal names. But certainly humans use them for gossip ;)
Language/dialect could also play a part. Humans form very large social groups that share a common language, so even if a culture comes up with a "unique" names (e.g. no special "name" words) in a local group, there will still likely be collisions with others in the wider group.
There's still a general rule to keep names unique within local groups, and it's weird to intentionally do otherwise (see Pete & Pete).
These animals might have "dialects" that only span a small number of individuals, so part of that "uniqueness" may just be dialect. I think you might see something similar in humans if you looked at small scale illiterate cultures in areas with a lot of linguistic diversity (e.g. Papua New Guinea)
My favorite surname detail is that in Iceland you surname is derived from your father. So Björk Guðmundsdóttir is the daugher (dóttir) of Guðmund. Her brother would have the last name Guðmundson. I'm not sure if any of the other nordic countries do the same. Russia put the patronymic in the middle name. Anyway is interesting in that it carries genealogy info without having a long established family name.
The patronymic is also frequently used - to address someone respectfully, you'd use their first name and patronymic, not their first-and-last name. (e.g., Dmitriy Alekseyevich Kuznetsov would typically be addressed as Dmitriy Alekseyevich in just about any non-informal context.)
Human names in an oral culture on the smaller family-and-friends scale seem like they'd be much more diverse, unique, and informal, with names that are more like nicknames than legal names. But certainly humans use them for gossip ;)