You are not incorrect in what you say, but there's still more to this story:
1) There is indeed an overarching umbrella for Marxism and identity politics: it's called critical theory. Since the 1930s, some revisionist Marxist academics have tried to expand what they saw as the 'method of Marx' to cultural concerns, and the resulting product is critical theory. These academics regard both (economic) Marxism and cultural theories to all be 'critical theories', as they focus on structural power and oppression using 'the method of Marx'.
2) A lot of identity politics is derived from the aforementioned tradition of critical theory, and its more postmodern offshoots like critical race theory and critical gender theory. Furthermore, the relativist epistemiology of postmodernism is sort of derived from Marxist beliefs too, specifically about how all 'superstructural' phenomenon are there to serve the ruling (oppressor) class. Ideas like these probably couldn't have formed without Marxism being there in the first place!
3) A lot of the identity politics language is also distinctly Marxian as a result. e.g. the 'appropriation' in cultural appropriation is arguably a Marxian usage. It is hence clear that identity politics is much closer to Marxism than liberalism in its worldview.
4) There are many self-identified Marxists who mix (economic) Marxism with critical theory in their activism nowadays. They even call old-school economic Marxists 'class reductionists' (the irony!). The two are very mixable because they are ultimately similar in structure, unlike e.g. how you can't mix Marxism with liberalism or conservatism. That's why, the debate between 'Marxist class reductionists' and 'identity intersectionalists' are more like heated arguments under one umbrella, akin to neoconservatives vs paleoconservatives (but they are still both conservatives).
5) Therefore, in conclusion, it is important to illustrate and explore the links between identity politics and Marxism, even if Marxism itself doesn't include identity politics.
