The Big Five in your quote are a perfect example: They have a significant heritable component, but for all factors heritability ranges from 40 to 60%, meaning 60% to 40% are explained by environment. Yet when listening to Peterson somehow the environmental factor doesn't seem to matter at all, which is clearly in contrast to the current state of research.
But just listen carefully to his more political talks. I sadly can't find it right now, but there was a mythical lecture about different cultures all having a variant of a tree of life. He simply lists a few cultures and then states that this list obviously can't be coincidence, ergo god exists.
He also routinely phrases things like "people high on consciencousness tend to be more conservative" in his public appearences. Which is true! The problem is in the context and the way he says it implies causation where there is no proof of any. Same thing for the gender differences.
Unrelated, but I personally really dislike his strawman arguments like his stance that postmodernism equals leftism, then criticising postmodernist theory. Practically all leftist/materialist theorists are directly opposed to postmodernists. There have been decades of bitter ivory tower battles between both factions.
Mind you I have no axe to grind here. I actually feel really sorry for the things the social media mob did to him. Just from watching his videos you can really see how bitter that experience made him in just a few months. The tragic thing is that in that original 2016 incidence, he actually was right! It really is futile to try to fight human tendency to categorize by changing the language for the categories. And there there is abundent evidence for that. (Amusingly the idea of changing reality trough language is actually a conservative one, represented by e.g. Heidegger.)