I got started in campus politics on a bathroom issue in the mid-ish 1960s and really didn't think we'd be back to such issues in the second decade of the 21st c. Anyway, I see that signs on bathrooms — who, uh, goes where — are back in the news, so here's some background from the last few rounds of argument.
Homo sapiens sapiens, "The wise, wise man," as we arrogantly call our species, is sexually dimorphic. That means that "the modal phenotype" is bi-modal: the folk you're likely to run into are usually XY sexual males or XX sexual females. There are other places on the curve, however, and even sexual expression gets complicated, although not often enough complicated enough that variations are, as a practical matter, difficult to accommodate. (This really should not be an issue for major debate.)
"Gender" is a term from grammar which, the lore on the subject tells us, got introduced into a legal brief to avoid repeating "sex," "sex," "sex." The usage got picked up and expanded as a handy way to talk about how people's sex gets culturally and socially manipulated in its expression in gender roles, yielding curves that are also usually bimodal — culture and society are powerful — but pretty complex: cultures and societies evolve a whole lot faster than species and go through periods of relative fluidity, and are always subject to The Iron Laws of Fashion, as the sainted Mark Twain saith in "Corn-Pone Opinions," even while stressing the power of social pressure.
So: The Great Bathroom Debates are indeed about who gets to piss next to whom, and with what plumbing, but more crucially about who gets the power to make such decisions and about hierarchy: Executive Toilets, Faculty Johns, Whites Only. In the couple dorms I've been in where decision was left to the residents, the matter was settled without fuss. In one for older folks, a woman who was a Mother Superior in her usual life just posted signs for MEN/WOMEN; in the other, U of Chicago undergrads just shared washrooms, with very strong social pressure on guys to put penises back in their pants and zip up before turning away from a urinal. Introduce parents into the decision-making, and there's another group going for the power to decide.
Oh, yeah: And then there's religion and archaic fears underlying religion. See Mary Douglas's Purity and Danger, especially on "The Abominations in Leviticus" and note more widespread views of Chaos as the breaking down of boundaries. And before you flatter yourself into thinking how far beyond such fears you've gone and how you can groove on boundary transgression and fluidity — watch Ridley Scott's ALIEN again, followed by the James Cameron and Gale Anne Hurd ALIENS, and look through H. R. Giger's Necronomicon. Part of the terror of the Alien and its works is the violation of boundaries starting with human flesh but including male/female, mechanical/organic. So for the passion of john debates, add legitimate issues of boundary policing and irrational fears — noting that fear, as such, is an emotion and not exactly rational even when totally justified and whenb, so to speak, a damn good idea. (Fear over bathroom issues: probably a bad idea.)