Currently, all villagers and players are either seen as male or female, with no in between. I propose that there should be an option for nonbinary villagers/players.

You are mixing gender/sexual orientation with sex. There are traits for that, as well as config options to turn gender off all together.

I can't find a nonbinary trait in any menu. Where is it?

nonbinary would not have an effect ingame, thus there is no separate trait. Only bisexual.