Okay, let's imagine a situation. In class:
"Yes, the lady in the back".
"Excuse me, Mr. Peterson. I prefer you call me 'gentleman' rather than 'lady'"
(Normal world:) "Sorry. The GENTLEMAN in the back"
(Petersons world:) "I REFUSE to pander to your Marxist ideology!" (Black helicopters swoop in and Jordan Peterson is taken to room 101).
From what I have heard of Peterson you have totally misrepresented his position. If a transgender person wants to be known as he or she he would have absolutely no problem with that. Neither would I.