Term similar to feminist but focused on unfair prejudices against *either* gender?

Egalitarianism works, though if speaking about socioeconomic factors (which always get tied in with gender inequality), usage may implicitly skew more to the side of economic egalitarianism (i.e. "the 99%").

You could specify further with "gender egalitarianism."

I would go with anti-sexist, defined as "One who is against or opposed to sexism; one who does not think that either sex is superior to the other; one who considers the sexes to be equal" (here).

Whether one don't want to see any extravagant claims of a gender but only a human being then it's logical to merely think about humanism. https://en.wikipedia.org/wiki/Humanism