"(a system that’s distinct from our diverse gender identities)" Please don't take this the wrong way, this is a genuine question. Is there an explaination somewhere of what purpose gender identities serve and what benefits that provides (ie why don't animals need them)? Aside from reproduction, I don't see how gender would really influence anything.
Gender is sort of a natural thing in human identities, even if it is a social construct. Intersex individuals are a pretty good piece of evidence of this. People born with ambiguous genitalia. Doctors often pick which one to go with and will do surgery to make it so. They often “get it wrong” and the individual will later feel they are the wrong gender WITHOUT necessarily knowing they were intersex at birth.
Gender can be a social construct and still have biological intrinsic factors behind it. It doesn’t need to arise from a purpose per se. it doesn’t NEED to have biological intrinsic factors either.