Holdovers from showing the case (function in the sentence) of pronouns with inflection, which has mostly vanished from English, but not completely. The "mostly" just means we get extra confusion from different cases having ended up with the same form (as in the two different meanings of "her" below).

"It was him -> it was her" -- both are direct objects (at least as you're using them here--once upon a time the correct English would have been "it was he -> it was she", since the verb "to be" is not a transitive verb, but that seems to have gone out of fashion).

"It is his -> it is hers" -- both are possessives, but they're not directly modifying anything (they refer to "it", but the verb is in between).

"It is his thing -> it is her thing" -- both are possessives, but now they are functioning as direct modifiers of "thing", which is a different usage.

I don't know if there is a fully specified algorithm for determining these things, and even if you got one that worked in English, I don't think it would map very easily to any other language, because English is such a hodgepodge of things cherry picked from various languages over the centuries.