Since I'm bored and know a thing or two about the subject, I'll chime in.
In this clip, Peterson (I believe unintentionally) misrepresents postmodernism as an ideological tool which is used to gain power by collapsing commonly held notions of race, gender, class, etc (the mistake, if not clear, being that he believes the purpose of postmodernism is to gain power). He then mistakenly bundles postmodernism with an unwieldy strawman of traditional Marxism—oppressed vs. oppressor—and sets fire to the whole project to cast light on his own supposedly more wholesome, unifying, humanistic, empirical psychoanalytic perspective of people and power, which is bullshit.
He then falsely equates postmodernism with identity politics, which he incorrectly defines as the ideology of those who insist "that the most important element of any student or any person for that matter is whatever racial, gender, or sex identity happens to be flavor of the month" instead of more accurately as an exploration of the tendency of people to align themselves with and promote those aforementioned and other limited ideological factions, which philosophers and even statesmen have been exploring for hundreds of years.
He then expresses an unironic authoritarian lament for authoritarian control of identity categories for which he wishes there were more simplistic guidelines: see his discussion about fractional blackness. He sees, in inversion, the trees for the forest.
Peterson then throws out a homily of questions about who, in the absence of "authority," is going to question, demarcate, catalog, and promote the "correct" solutions to these problems. This shit goes on and on and on, and so that I won't bore you any longer, I'll get to the point.
Peterson confuses contemporary outrage for postmodernism. He sees in uninformed keening the foghorn of a postmodern boogeyman. His outlook seems to be based on viral videos (and, admittedly, experience) of misguided university students screaming for attention. From this he's constructed perverted misinterpretations of their youthful ignorance as stonewalls of postmodernism.
Postmodernism is not ideology. It's not authoritarian. It's not identity politics. It negates all of that, and it's not anything that Peterson or any mission-driven clown says it is. It's a mode of questioning how and why we believe what we do. Its purpose is to probe the assumptions we unwittingly bestow on leaders, institutions, media, and even seemingly minor shit such as "authentic" Mexican recipes. It's a current by which you can bypass the crashing waves and bottomless depths of traditional ideologies. It embraces the type of critical questioning Peterson claims it doesn't, and it employs the type of skepticism of movements and their individuals for which Peterson squawks and burps.
The ultimate irony is that Peterson's goofy little talk in this clip likely wouldn't be possible without postmodernism.