The thing is there isn't a direct crossover from race to sex. You just can't do find/replace with the story. This is because:-
Sex is something that is a part of your body and DNA. There are different hormone balances in your body depending on which sex you are, (and your age of course)
Race is something that is assigned to you. In this culture, whiteness is seen as the default, everyone else is "other" and given second class status. You don't feel white or black, you are treated that way by everyone else. Sometimes there are little perks like being seen as cool or smart but mostly it is a de-buff because it's a system of white supremacy we are in. So, there are some white people who get all the benefits and privileges of their whiteness who still want to access the little perks as well. There's a spectrum, trans-racials like Rachel Dolezal are down one extreme. This instagrammer is over that end but not quite as extreme.
Black and other people of colour, on the other hand, don't have that option. "Passing" is the closest analogue but it's not like it is for white people. White people can quit culturally appropriating/blackface cosplaying whenever they want and full white privilege resumes. The worst that could happen is they are made fun of on social media. Being discovered "passing" was life threatening. Most of the time, it's just not an option.
Someone is black because everyone says they are black and everyone says it because white supremacy needs something to be supreme over.
Gender is also socially constructed, and in the patriarchal system we live in females are assigned to a second class status too. So far, it seems similar to race, right?
We are all socialised to see men, women, girls and boys as having certain characteristics and levels of privilege. Just like we are socialised into white supremacy.
But even in an existing non-patriarchal society or even in a thought experiment society with no gender socialisation, it still feels a certain way to be male or to be female. It's the hormones, mostly. Maybe even brain structure (but I believe there is slim to none evidence for that) Definitely our physicality. We go about life in a different way because we have different bodies.
For me, it seems more than possible that a person would feel socialised into the "wrong" role. Especially in a sick patriarchal culture such as ours.
But transsexual and transgender people don't exist because of patriarchy/male supremacy. They have existed throughout history, across cultures.
What happens next is culture dependent. Some cultures said "fine, just cross dress and we'll all treat you like the opposite gender", some said "ok, so you're neither then, here is a special role for you", some said "nope, that's not a thing, conform to your gender or we'll hurt you" and ours right now is changing from saying "we can change your body to as close to the opposite sex as possible but you better not be discovered" to something new.
That's not to say there aren't people claiming to be trans when they are just cosplaying tourists, and more like these people trying on blackface for instagram likes. You can tell them from the others because they go running back to male privilege when it suits them/gets them into the nicer facilities in their golf club.
But it isn't directly equivalent. Just similar.
That's what I think anyway.