There are at least two different classes of objection to gender-affirming care: (a) objection to the care itself; and (b) objection to an ideology seen as implied by the care. These are clearly distinguishable, but they are often conflated in the minds of objectors.
"Gender-affirming" is, perhaps, an unfortunate term. Feminists have long supported the idea that gender is a mere social construct, and many in the transgender community have unfortunately adopted this idea, perhaps because "the enemy of my enemy is my friend." But the social construct theory sees gender as artificial, having no objective "essence," and is therefore inherently delegitimizing to transgender people.
The idea of gender that is supportive to transgender people is that gender is an as-yet theoretical, objective, intangible quality on the male-female spectrum that is usually confluent with sex but it potentially divergent therefrom. Focus for a moment on the word "theoretical": there is no proof that gender, as here defined, even exists. Perhaps it does, perhaps it doesn't.
So why to transgender people subscribe to it? Because it's an explanation for why they feel the way they do. Trans people, like all people, seek meaning in the conflicts of their lives; and the idea of gender gives their conflicts meaning. Does the status of "belief" mean "untrue"? Of course not. We can't prove it exists, but it may. We just don't know.
Thus, medicine does not provide "gender-affirming" care based on the idea that gender is real. Medicine, and science in general, doesn't know whether it's real or not. No; the reason that medicine provides "gender-affirming" care is that it relieves suffering, when nothing else does.
Is it dangerous? You can certainly say so. Dr. Marci Bowers, herself a trans woman and a prominent transgender surgeon and figure in the trans community, has recently come out saying that there are permanent and negative consequences to some medical treatments given to minors. These negatives must be weighed, by parents in conjunction with competent medical advice, against the positives to be gained by the treatment. Medicine by legislation is like performing surgery with a machete: you may succeed in cutting off things that need cutting off, but you will probably end up cutting off things that don't, as well. And you may miss altogether.
I don't know if Peterson is conflating these classes of objection, or simply objecting to all medical care of this nature on the basis of medicine alone. Is he simply saying that the idea is a lie that, on balance, care is never more helpful than not? or is he addressing the proposition of "gender" as the lie? or is it both? The first is a medical question; and he can speak with some professional authority to the psychological side of things. That doesn't mean he's right, of course, but it does mean his comments in that category have a certain weight. The second is a metaphysical question; and, here, he has only such authority as he is given by his listeners.
Transgender medical care for minors is sometimes compared to the work of Dr. Josef Mengele during WW2, who, for example, injected blue dye into the eyes of Jewish children, blinding them, to see if they could be made to look more Aryan. In so doing, they completely ignore the element of motive.
I don't think it's unreasonable to make the process for providing care for minors more rigorous; but outlawing altogether medical treatment -- especially medical treatment that the majority of the medical community seems to agree with -- seems reactionary and heavy-handed.
Some people behave or support authoritarian action because they're tyrants at heart, while others do it because they're frightened of what will happen if they don't. Sometimes you see something that strikes you as so evil by its very nature that your only course of action seems to be to stop it, by whatever means are necessary. This is only human; and I don't doubt that there are many ordinary citizens, and perhaps some politicians, who do feel this way. That's not to defend their actions; but it does speak to their motivation, and, as we all know (or should), there is no criminality without criminal motivation.
All this furor is tragic, because those most affected are caught in the middle and must depend on others to speak for them. Neither side in this conflict is blameless; but perhaps analysis of causes is best left for another day.