> The subjective experience is merely an identity based on cultural stereotypes of "man-ness" and "woman-ness".
No, it runs much deeper than that. Gender roles are cross-cultural, and fundamentally rooted in the fact that females can get pregnant and males can't. That's why the stereotypes exist, because they are rooted in biological facts. But those facts aren't absolute. Most people who have male genitalia also have certain behavioral predispositions which we associate with "being a man." But not all people with male genitalia have those behavioral predispositions, and for those who don't there is no reason not to treat them according to how they feel rather than what happens to be between their legs. (For starters, if you want to cast your lot with "biological facts" and treat people according to what happens to be between their legs, then you have to deal with the fact that there are natural hermaphrodites.)
> If I decide to wear a fancy hat and call myself King of the World, should I expect everyone to call me that?
Sure, why not?
https://en.wikipedia.org/wiki/Emperor_Norton
> people can be wrong about themselves
They can't be wrong about their own subjective experiences. An individual is the ultimate authority on that by definition.