The Marxist pejorative “vulgar dialectics” refers to the rigid, mechanical formula of Marxist-Leninist “Diamat”, the official philosophy of the Soviet Union etc. I don’t think the quotation is targeting only that.
The problem with the quotation is it confuses different kinds of contradiction.
As I said a couple of months ago:
But while this sounds pretty good, it’s not clear what it means.
Your quotation confuses (a) a critique of formal logic’s tendency to hypostasize or resolve, with (b) a rejection of formal logic as such. Formal logic is fine as far as it goes, but its abstractions can obscure the movement, interdependencies, and antagonisms of reality.
There are “subjective thought-errors” (Adorno), and dialectical philosophy ought to avoid them as much as any other philosophy does—but there are objective contradictions too. These cannot be reduced to “antagonisms”, “conflicts”, “tensions”, and so on. I’ll try to explain why with an example…
Exchange in the market both is and is not an exchange of equivalents: commodities are actually made and treated as equivalent, but since they are qualitatively different, they are also not equivalent. This seems easily resolved: commodities are equivalent in the economic context, but not in another.
But! The crucial point is that “in the economic context” is not just a perspective we can choose to adopt or not. Economic equivalence is an abstraction, but a real one: it governs what people actually do. And what it governs is not a mere aspect of things but the things themselves, which are really exchanged as interchangeable while really not being so. This is not bad thinking to be sorted out with finer distinctions.
That’s what my earlier quoted line was getting at. We keep to non-contradiction in thought, and that’s why a real contradiction like this shows up as a contradiction in the first place.
Not all dialectics is rationalist (the claim I made that reality need not obey the rules of thought is the opposite of rationalism), but I concede it tends to rely on essences. The difference is that Marx and Adorno historicized essence: it is not eternal, but the internal necessities of a particular form of society, which could itself be otherwise.