It seems to me it's not so much that Jordan Peterson must use certain words, but rather that he should avoid using words that are offensive to the people he is talking to. In avoiding offensive terms of address there is not necessarily a compulsion to use a particular alternative.
In many situations we can get away with not using pronouns, instead simply using names, gestures, or even "you at the back". I very much doubt anyone is getting into trouble simply for avoiding pronouns.
The problem probably begins when someone asks to be addressed in a certain way and this guy says no, and what's more I'll address you instead in precisely the way that you find offensive.
So it's not about being forced to use certain words. It's about Jordan Peterson insisting on his "right" to use words to describe other people that they find offensive.
If we agree that black people shouldn't be addressed using the N word or women shouldn't be addressed with sexists names like, pet, doll, honey, or whatever. Then it seems pretty reasonable that transgender people be allowed to insist that they not be addressed using language that they find distressing. I know I would find it unpleasant to be addressed as a woman. Jordan Peterson claims he wouldn't care, but that just makes him either pretty strange or a liar as far I can see. In either case it doesn't matter, most people would find it distressing to be addressed by the wrong gender in spite of pleas to discontinue.
The onus is on Jordan Peterson to explain why it would be wrong to address a black person with the N word, or a woman as "doll" or other sexist terms, but that it's okay, in fact it is his "right", to address someone as he and him and his, who does not feel themselves to be a man.
We have a president of the United States who is intimidating the press, encouraged violence in the campaign, is stigmatising minority groups, and threatened to put political opponents in jail. Any sense of proportion at this moment would allocate no time whatever to mister storm in a teacup Peterson.
Interesting bohm points out he's making money and gaining a following by means of this storm in a teacup.