As far as I can tell, Professor Peterson is a competent clinical psychologist and researcher in the domain of the Big Five personality traits. He may or may not be a reasonably good self-help guru. The people who like that kind of thing like that kind of thing. His broader cultural and social pronouncements are a bit nonsense. He is not particularly important, nor is he dangerous, nor is he worth all the fuss that's been made. I was inspired to deal with him only because the recent BBC interview was really bad; the interviewer was incompetent in both failing to listen to and address anything he said and taking a moralistic tone of calling him out for being bad without first addressing whether he was wrong. Also, while I annoyed with some ideologies that Peterson dislikes, his reasons are wrong and his worries are a bit histrionic.
Peterson worries about ‘identity politics’ and ‘political correctness’ and while there are trends and subgroups in that area I dislike, his likening them to Bolshevism and scare mongering about a totalitarian regime is essentially nonsense. The real danger of the more authoritarian elements on the left is their own incompetence; adoption of the thoughts and practices of the ‘critical left’, to use Rorty's term, seems likely to slow progress, dissipate energy, and strengthen reactionary forces.
More interesting is his railing against post-modernism, since everything indicates to me that Professor Peterson is essentially a moderate conservative postmodern himself, following the same ideas but with a more popular framing than the rather dry and convoluted prose more common in the humanities departments he criticizes. Where the academic postmoderns were inflected by horror over World War II and the Holocaust, Peterson goes the same direction as a result of his fascination with the regime Stalin. The rough shape of his reasoning is plain in even casual comments, such as his discussion with Sam Harris where he subsumes instrumental reason (without using the word) to his weird and rather arbitrary ‘evolutionary’ teleology. The best thing that could happen to Professor Peterson would be if a few works of Steve Pinker and Peter Singer fell on top of him one day, as they both reject the tabula rasa and accept much of what biology and evolution suggest, but rather than bowing to and fetishizing it as Peterson does, they take it as a starting point for rational analysis about how to organize civilization to improve equality and human flourishing. I decided to read one of his books to see if my impression of him as fundamentally a vulgar postmodern was correct.
So, let's take him on his own terms. Peterson, it is safe to say, thinks Gender is much more important than I do. He thinks there are Appropriate Roles and many of those come from Inbuilt Facts of Nature. As far as I can tell he thinks there are good masculine identities for male humans and good feminine identities for female humans and that society should affirm them and encourage maturation into them. He seems to think that failing to affirm masculinity is a bad thing that can be done to people.
In the first chapter of his new book, he spends a good bit of time talking about bullying. About the psychological and physiological harm one can do by attacking someone's status and worth. As far as I have read, he doesn't mention it explicitly, but given his other beliefs I would be very surprised if he would not consider things like aggressive verbal feminization, taunts of ‘girly-boy’, and the mocking address of a young man with female pronouns or a stereotypical feminine name examples of harmful bullying. It's a direct assault on status regarding facets of identity he considers important defining.
Whatever one thinks of hate speech generally, I think one can make a good case that in institutions that people are required to interact with to gain credentials, learning, or for other purposes of their livelihood, bullying is generally not a good thing. Peers or co-workers actively harming each other is bad. Those in positions of power such as professors or managers doing so is even worse. For him to specifically misgender people even when informed that he is doing so and has been asked not to would constitute some level of bullying. It's a conscious decision on his part to reject another person's status in an aspect of life that is very important to them. (Transitioning is hard. People don't do it in fits of whimsy.) Note that this has nothing to do with a limitation on inquiry or investigation. He can say anything he wants about his theories of gender, transitioning, or what have you, even if his theories are false. There is a difference between essentially content-free insults in school or the workplace and advocating ideas, even ones which are almost certainly false and offend many people.
The question one should be asking Professor Peterson is not what gives him the ‘authority’ to say this or that, but why he is spending effort defending what looks, by his principles, to be bullying in school or the workplace. What greater social good does he protect?
When you look past all the malarky about Bolshevism and the Gulag Archipelago, his concern is compelled speech. He brings up zie and ey and other newfangled pronouns and doesn't think he or others should be forced to deal with them. I have some limited sympathy for this. One could reasonably argue that every new pronoun puts a cognitive burden on people, and that one while one may expect ones friends and loved ones to track and use innovative pronouns, it is unreasonable in the business environment. I'd be tempted to counter-argue that the number of pronouns in actual use is not actually likely to balloon that much, and that most competent managers will probably want to learn that kind of information anyway to make employees or students feel like they matter and inspire them to perform well.
Nevertheless, let us grant this objection. Legal scholars who have commented on Bill C-16 have stated that one would not be required to positively affirm an other's gender, merely to avoid denigrating or denying it. This sounds very little like compelled speech, and very much like expectations that people not use racial slurs in the workplace.
There are lots of ways to do that. Circumlocutions abound and English is a language of splendid variety and expression. The common method is singular ‘they’. Professor Peterson does not like singular ‘they’. It confuses plurals, it sounds weird to him…
There is an appropriate response to this, and I believe it is “Stop being so precious and sort yourself out, bucko. You ever go to China? You know their language lacks both gender and number marking on all the pronouns? You can explicitly signal them, but by default they're ambiguous. Ever speak English? You know we don't mark number on the second person pronoun, right? So we'll get ‘they’ and ‘they all’ the way we have ‘you’ and ‘you all’ now. You worry it sounds funny? So what? Do you want to the kind of person who hurts people in a way your own philosophy suggests is harmful because you don't want to sound funny? What kind of principle of ordered society is that? Why don't you pull yourself together and go clean up your room?”
In all honesty I wouldn't be surprised if the only reason he sticks to this whole pronoun thing is that it brought him notoriety in the first place and, at least subconsciously, he wants to keep that fame.
Peterson's other comment, that whether he would honor someone's request depended on how they ask, is itself slightly troubling. He is a clinical psychologist. He knows the effects of stress and how it can affect people's behavior. He writes about them. He knows it can bring out the worst in people and that being mistreated can be a cause of chronic stress. While I am not a fan of targeting protected groups specifically in law, I do think we should keep in mind which people have been subject to discrimination and poor treatment in recent history and to let that knowledge influence our reasoning. It is well-known that trans and non-binary people face mockery and challenges to that portion of their identity. Since Professor Peterson is the professor, the adult in the room, and the one handing out life advice on how to grow up, it is really inappropriate for him to act as if there is some decision to be made as to whether he will be an adult and treat people with the common baseline of decency simply because they asked him to in a rude or way.