Like many things, gender is a mix of both. It's an immutable core wrapped in various social and cultural markers we've come to associate with "masculine" and "feminine". And of course the range of possible cores is more of a continuum than a simple binary flag (even though we like sorting all the arbitrary markers into two neat piles as if that really means anything).

Considering we still largely refused to acknowledge something as intuitively obvious[0] as non-right-handedness less than two generations ago, I wonder how long it will take until we've learned to deal with the fact that gender (and sexuality) is as idiosyncratic as nearly everything else about an individual's identity.

[0]: Seriously. People generally have two hands. Why on Earth should we assume that there's something special about the one that is dominant for most people? Good luck explaining gender and sexuality to a society that can't even imagine not everybody is strictly right-handed.