He does address identity politics in a nuanced way throughout his work, he views it as the hysterical response which is also a necessary/ fundamental moment in articulating a critique of ideology and hegemony, it's merely that this, the proliferation of the multiplicity of identities, is not in itself revolutionary. He absolutely believes trans people have the right to exist, housing, work, have full political agency, families, friends... His whole point is that these things should not hinge on whether identity of the other is merely tolerated, but on the basis of a shared emancipatory project which excepts no one. I just think he principally refuses to fetishize transgender identity (which can appear callous and be weaponized through reactionary framing).

I think a good feminist interlocutor from the same Lacanian frame as Zizek is Mari Ruti if you want to check her out. She offers a ton engagement with the notion of identity and most of her work I've read has also woven in very valid critiques of Zizek (and while these are grounded in a more individualist frame, she writes in a much more accessible manner). I get that Lacanian discourses around the phallus, castration, hysteria and whatnot can sound hyperpatriarchal, but the core implications are actually radically egalitarian if you grasp them correctly. Castration is the precondition of subjectivity for all of us.

Also the thumbnail above is obviously inflammatory relative to the content of the video, come on