A "woman" is someone who identifies with the corresponding social construct that is defined by cultural expectations and roles of varying rigidity and status depending on culture.

Nope. "Woman" is the term used for an adult female of the human species. Female, in turn, is a reproductive category common to those animal and plant species which perpetuate themselves by sexual, as opposed to asexual, means. As such, both it and the human-specific "woman" long predate ANY social constructs. (Indeed, women existed long before the word for them did, or language itself for that matter.)

Moreover, the "social construct" to which you refer-- the female gender-role-- is not based on how one identifies at all. It's based entirely on one's biological sex, and applies automatically as soon as this is perceived. Which it almost invariably is, since (like all other animal species) we are equipped by nature to recognize our own sexes.

Finally, while the female gender-role does vary by culture in many respects, one thing about is universal: always meaning "inferior". So, in order to "identify" with it, one must necessarily embrace inferiority-- subordination to male people-- and all that this entails: being paid less, treated like an eternal servant, beaten, raped, etc. How many women truly accept such status? And how can men who "identify" as women ever be said to do so? Few if any actually "pass", so they are instantly recognized as men, which means that the male gender-role automatically applies to them regardless of any personal ladyfeelz^TM. Thus, they are only as "inferior" as they choose to be. It's just a game to them; for women, who are assigned this role with no opt-out, it's another matter altogether.