The technical difference between modernity/classical industrial capitalism and postmodernity/late capitalism isn’t Marx’s, but that of later Marxists / cultural materialists.
It might be summed up as follows: in modernity / classical industrial capitalism, we can become aware of our own alienation from our own labour, and of the fissures between ideology (or the “superstructure” of culture, discourse, ideological positioning etc., in Althusserisn terms), and the material base of our exploitation underneath. In postmodernity/late capitalism, we become unable to tell the difference between materiality and ideology, and in fact late capitalist culture (postmodernity) explicitly works to upend our ability to tell ideology from material exploitation, and to turn our alienation into a feature of the culture that is not just obfuscated, but celebrated. People become unable to tell fact from fiction, representation from reality, ideology from materiality: and far from this being a concern, it’s experienced as a kind of enraptured absorption into commodification, ideology, abjection of all kinds - a sort of joyful wallowing in alienation as identity rather than seeing alienation as a indication that something is wrong with culture.
You can see here how Marxism is diametrically opposed to both late capitalism/postmodernity, and to the kind of individualist identity politics that it celebrates/wallows in.
The current formulation of identity politics, especially as it emerged in the US, came it of a rejection of Marxism and an embracing of a kind of activist version of neoliberalism. Marxism in the US has had almost no purchase in intellectual terms, aside from little pockets at Berkeley in the 70s and the New School in the 80s.
There’s no indigenous Marxist tradition in the US; but there are strong traditions of civil rights discourses that tend to make claims of separatism rather than universalism, and for which “identity” allows for a claim to Constitutional rights in a way that never was quite the case in the European tradition. These forms of civil rights discourses tend towards being fully complicit in US commodity-capitalism, and in fact often thrive on rejecting Marxist or socialist social thought.
IMO Peterson’s lazy reliance on tired old pseudo-McCarthyite tropes about Marxists and socialism is a huge gaping flaw in his thinking, which means above all that he isn’t interested in exploring the ways in which monetary, capitalist and state interests prop up transgender ideology in North America in particular.