For the conferences that I have been to in the last few years, I'd say that the understanding that is no direct biological basis for the Big 5 is relatively well-accepted in the research community. There are always (for good reasons) gaps between what is accepted in research, and what is accepted in general community. But as an academic (?) I'd expect Peterson to be up on the research. Personality factors such as the Big 5 can only be reifications based on frequency of endorsement correlations of items, which may not be well specified, for what exactly, the item is meant to evaluate (I dare not use the word measure, there). What the factors evidence is something perhaps of the structure of language, and shared meanings as evidenced in language, much more so than anything intrinsic to men, women, or any one human (except perhaps, the original researcher who came up with the item, although we never require that researcher to justify the item construction).
My encouragement is to consider the kinds of information that Peterson is relying on, in his formulations of characters, gender qualities, and personality, particularly. His claims have brought me to reflect on the meaning of scientific respectability - particularly in regards to the Big 5, which yes, has widespread use in psychology, but which also, has no biological basis to its formulation.
That makes me curious - he claims biological determinism for hierarchies, but makes room for social construction, in his use of personality theory (the Big 5 - there is nowhere that the Big 5 comes from, except for frequencies of endorsements of questionnaire items, by people, even, many people, through time).
Social constructionism has roots typically traced to postmodernism - I just think it is an interesting kind of double standard. He seems to play on or with a few double standards - the action that I understand he has taken regarding the legality of preferred pronouns (calling for the shut down of those who seek to shut down other's freedoms of speech) seems to me to be another double standard.
When it comes to science - would this seem to be a good basis for knowledge, is a question for my research - are we okay with utilising constructs which are created from shared understandings, rather than some deeper sense of what a psychological phenomena is.
Peterson has a whole body of work that builds on the Big 5, you can check out his Google citations page for this.
None of it really seems to question the Big 5 - which I would hope, good science would do, or at least, would be interested in. There are good reasons (and respected voices in the field) who challenge the Big 5) - I can give links for these authors and papers, if they're helpful.
The question of scientific respectability in this regard extends to any set of test or questionnaire items formulated by factor analytical methods that have only made use of statistical analyses, in their claims about the existence of things like openness, or agreeableness, or conscientiousness.
If I think about my Tim Minchin-inspired skepticism, I become curious about what sits behind the things that the psychologist in question, talks about. Peterson refers to the Big 5 factors as real existent entities, that move people about in the world, in certain ways (having causal efficacy). There is no biological basis on which a connection between a Big 5 factor, and a single human person, can be claimed.
I lose sense of what science means, in this regard. I'm doubly baffled, because Peterson insists on biological determinism when it comes to social hierarchies, but the antithesis of biological determinism when it comes to personalities - and this is notionally actually in his field of expertise. Psychology."