In terms of prevailing theory, they aren't actually that different. Gender is seen increasingly as a performance, especially since Butler's Gender Trouble. Actually, 'born this way' is now largely a questionable way of getting liberal legal theory to grudgingly give rights to LGBT people, because liberal legal theory really does not do well with anything that isn't obviously a liberal issue - and the denial of rights based on an accident of birth is obviously a legal issue. There was an article recently on Hubski about 'born this way' actually: http://aeon.co/magazine/society/why-born-gay-is-a-dangerous-idea/
I'm doing a gender studies masters, most theorists - particularly in queer, poststructuralist traditions - don't see sexuality as innate. They don't see it as a choice, either, of course. Gender and sexuality are part of the process of being constituted in discourse, and discourses about gender and sex are inseperable. Even as a straight cis mail, you were constituted in gendered discourse just as much as a trans lesbian. You just don't have to agitate for your rights as a straight cis man politically.
Language isn't innate, either. But learning a language is an inevitable part of growing up and growing a conscious. And we all speak SOME language, even if some languages are more associated with oppression than others.
Plus, less theoretically, political alliances are important. Having similar experiences and problems is more than enough for a political alliance.
By the way, if you want a really fascinating look at how discourses of gender and sexuality can be constructed radically differently from ours that doesn't involve a lot of complicated theory, read volumes 2 and 3 of Foucault's HIstory of Sexuality.