Specifying Pronouns as He/Him;She/Her;They/Them

It is becoming more common for people to explicitly state the pronouns to be used in addressing themselves: he/him, she/her, or they/them. For example, a name tag for a conference might read:

Cory Lopez
Some Company
She/Her

Usage seems to have settled on the singular they and them as gender neutral pronouns. Alternatives that were once widespread in niche communities, such as sie/hir, seem to have died out.

Whether or not one agrees with this language shift, the pairing between subject and object case is fixed. It's difficult to imagine someone who uses "he" as his subject case using anything but "him" as object case, and it's even harder to imagine that such a person would succeed in getting people to use a non-obvious collection of personal pronouns. (And you may see what I did -- flat out assumed that "he" and "his" automatically go together.)

What are the roots of the practice of presenting subject and object case pronouns as opposed to just subject case, subject and possessive, or some other combination?

(Note: I am not interested in a discussion of the gender politics and cultural shifts involved, just the mechanics of how the current representation was chosen.)

To my knowledge, it's a convention that persists because the form itself has acquired shared meaning.

I don't know all of the details, but we can look at common patterns of usage:

1. Explanatory usefulness

   It's true that associations between subject + object forms tend to be fixed. However, in a situation where some people might be using the rarer "new" pronouns, it's quite possible that not everyone will know them.

   Writing "ze/zir" on a name badge helps people who haven't encountered ze before to understand the expected usage without having to interrupt the conversation to ask quite as much (they may still need to gently inquire about the possessive form).

   Another contributing factor is that there may be multiple competing conventions. E.g. ze has appeared both as "ze/zir" and "ze/hir," according to Wikipedia. Providing two cases doesn't disambiguate between all of them perfectly, but it's a compromise between clarity and brevity.

2. Through usage, the form itself acquires meaning

   Once you've got a stylistic convention in widespread use among a community of users, it carries its own bit of meaning. To an audience used to trans-inclusive spaces, seeing pronoun/pronoun in someone's name badge or social media bio immediately registers as "these are my pronouns." "He/him" is more immediately recognizable than just writing "he" somewhere in a small block of text all by itself; it's also shorter than saying "my pronoun is he."

3. A permutation: flexible pronouns

   Because the convention itself is recognizable, the specific nominative/accusative pairing isn't particularly important. Sometimes what you'll see instead is that people who accept multiple pronouns (e.g. both a gendered pronoun and a gender-neutral pronoun) will join two nominative-case pronouns with a slash, such as "she/they."

   In my experience, people intuitively register the difference between the two-cases and two-different-pronouns forms, for the most part. The underlying idea is the same and familiarity with pronoun inflection fills in the rest.