Conservatives keep trying to connect identitarianism and privilege theory to "cultural Marxism", a name I hate because "cultural Marxism" is to Marxism as the Reverend Moon's Unification Church is to Christianity. "Cultural Marxism" should be called something like "Frankfurt Schoolism".
But I digress before I begin. Onward:
Privilege theory in the US comes from the Ivy League via two academics, Derrick Bell, father of Critical Race Theory, and Kimberlé Crenshaw, coiner of "intersectionality". I haven't been able to find anything linking either to socialism or Marxism. Bell is on record saying he didn't read Marx, and there's nothing to suggest he was lying.
But it's true there are socialists who fall for identitarian theories without noticing their origin or how, as David Harvey noted, those theories serve neoliberalism. These socialists may well come from the Frankfurt School—I leave tracing that connection to people who care more about academic Marxism than I do.
My concern here is solely with socialists who question privilege theory. The quick googling brings up several pieces from the last couple of years that are worth reading:
On privilege theory and intersectionality by Camila Bassi.
Privilege theory: a political impasse? by Camila Bassi.
Is there a white skin privilege? by Bill Mullen.
An identitarian socialist's response to Mullen: White Skin Privilege and Marxism? by Brian Kwoba.
And a response to Kwoba: Towards a Marxist critique of ‘privilege theory’ by Tad Tietze.
What’s wrong with privilege theory? by Esme Choonara and Yuri Prasad, who found the perfect Marxist answer to people like Kwoba who claim, "To many race-conscious people, the idea that white people, because of their race, receive a whole host of unearned advantages and privileges is obvious." Choonara and Prasad quote Marx's observation: “All science would be superfluous if the outward appearance and the essence of things directly coincided.”
And two older favorites of mine:
Race, class, and "whiteness theory" by Sharon Smith.
The limits of antiracism by Adolph Reed Jr.