Depends on your culture. The Diné people have four genders (feminine man, masculine man, feminine woman, masculine woman), hijra and fa'afafine are feminine male genders, in Inuit culture a sipiniq is someone who fulfills a man's role with female genitalia.

In mesopotamia, mesoamerica, and the Indus valley there were people of unclear sex who fulfilled genders beyond men and women.

A neologism from pan indigenous culture is Two Spirit.

These concepts have existed for a loooong time, however they were quite rare in western countries. As westerners, we're being exposed to them now, but they are hardly novel on the world stage.

It's interesting though that all those cultures came up with distinct terms for trans identities. I suspect there'd be a lot less pushback if this was tried with English instead of overloading the meaning of pre-existing terms.

Probably true!

These are nice terms but they're heavily rooted in social constructs and don't really answer the OPS question which was tying biological sex to well... biology and in this particular case gametes/chromosomes.

You will never get an answer. Biological sex, gender, gender identity, gender expression and even gender roles are thrown into a pile to play 5D chess.

Fact remains, there is no third sex. Intersex is not a third sex and sexual anomalies also do not introduce a third sex.