Getting rid of names and pronouns is an interesting thought to design a fictional society. Let's walk through the stages of human communication (not scientifically backed here, working from observation):
1. Announcing your presence and intent to communicate
This is where names and descriptors are seemingly needed. On the most prototypical level, you'd have a small group of people, each of whom has some sort of descriptor assigned to them that makes them unique. This is typically a verbal descriptor, a name, like "John". Now, taking away names and their importance, possibly because the society places great value upon names being a private or inappropriate matter, there are other ways to adress someone.
a. Unspecifically depending on need / Temporary names
"The being able to accomplish this task shall do it now."
"The one who spoke to me yesterday was quite handsome"
These are temporary descriptors which apply to a certain situation. They still outline a more or less specific person (the first example being less, the second example more specific) and might quickly mutate to become names. This is probably how naming started, after all. While it is workable, I think this approach very quickly loses its non-commital nature and becomes commonplace once certain characteristics about persons are becoming universal descriptors ("John the Blacksmith").
b. Physical form of address
Distancing oneself from verbal means, a society independent from names might well be more physically inclined. Eye and physical contact may also be used to mark a person. This creates a very immediate, temporary association and basis for communication which does not bind either party more than the other party. However, this could be highly impractical.
c. Group characteristics
To avoid filtering a specific person by name, society might have developed in such a way so as to compose itself in social situations only of a singular representative of each group. For example, if these groups are Green, Red and Blue, any social assembly would only consist of a member or closely tied party of Green Red and Blue, but not of several parties of the same group.
A variant of this could be seen in titular and clan names, which are very unspecific to the individual but highly specific to the group, filtering out the ones possibly addressed. If only one member of the Clan of the Iulii was in the room, calling them Iulius would be appropriate in ancient Rome, as they represented their clan at that point.
2. Talking about others
This is where things get tricky. How do you speak about others if you can not individually describe them? Approach 1a still applies, but 1b is already a problem when you, for example, want to tell the police who just robbed you. 1c I could imagine, but only if societal rules allowed for a certain recording and regulation. For example "Blue attacked me" can either be understood as "The Blue in the room at the time attacked me" or "The entity Blue as a whole attacked me".
3. Societal Implications
Removing the idea of individual distinction seemingly points at a "communist hive-mind" society which simply is a fancy way to say "anarchy". I would argue that this is oversimplifying the matter. The main thing it points to is a lack of long-term focus, an emphasis on the immediate. If you have no name, you need no one to carry it on. You can seek no fame because you are not distinct from the rest. You can however distinguish yourself through your immediate deeds - that is, the things you are doing RIGHT NOW, because the second you stop doing them, the identity of the one who did them will be forgotten.
This raises interesting implications about the productivity of such a society, and its overall goals. One the one hand, if there is nothing to achieve on the long run, how does the human mind handle working at all? On the other hand, if working right now is the only way to achieve anything, how can you justify not working right now?
4. The "Why"
This is probably the hardest part of the question. Why would such a society even exist? Natural phenomena that make a short-term, highly reactive system more efficient? Overcompensation for past developments? Evolutionary considerations? Ideology? Many possibilities could converge to create an explanation.
PS: I smell amazing potential for a short story at the very least. If you weren't already planning to write it, I certainly would
