Before we even get to our big question, we need to clear preliminary underbrush aside. English does not really have a word for the small room furnished with a water closet (incorrectly called a toilet, a word that properly denotes not an object but the process or activity involving a hairbrush, innumerable lotions and unctions, a dressing gown, a mirror, and the like) and lavatory (semi-correctly called a washbasin and incorrectly called a sink). Perhaps because there is no possible way to bathe in it, we call it a bathroom or even more strangely, a bath. Other languages known to me are awash in similar fogs of euphemism and indirection here. Spanish-speaking countries often have these marked for caballeros (but I have never seen a hitching rail), and none for peatóns.
One way or another, we get to it, and find equipment for one person at a time, marked graphically or verbally for either men or women. Why? what modesty is offended by using one of these, alone, after someone of the opposite sex has done so, modesty that does not demand hotel rooms be so segregated? Why should someone be standing around waiting for the one on the left when the other is empty? But this is not a language question.
We may also find a sign that reads Unisex. Ah, and now we come to my question: why is a uniperson, disex (or multigender) bathroom, labelled with a madeup word that is wrong both ways, with no fastidious or euphemistic result?
Update 13/VIII/13: anyone who reads this far (and I’m humbled by seeing more than 30 comments!) deserves more and better lunacy. So with the thinnest possible connection, I nominate (i) Jonathan Miller’s immortal bit (by the way, the complete BtF is here, woo hoo!) and (ii) a Wikipedia article I never in my wildest fevered dreams imagined could exist, let alone extend beyond about fifty words.
It is a wondrous world, in so many ways