I think they're taking the piss out of the earnest "me me me"-ness of identity politics and are just using trans as a handy issue to do it with. (They may be trying to sneak in a dig at transgenderism as well, or they may view it as fundamentally different from the things they're taking the piss out. It isn't clear which.)
People who grow up with horrendous sex dysphoria and who truly can get relief no other way than by transitioning are not helped at all by trendy student-age identity politickers of the sort this sketch is taking the piss out of.