The short answer:
Nobody like to play the Oppression Olympics.
A longer (but still incomplete) answer:
Authoritarian structures, congealed in the state and other institutional hierarchies, permeate all class-based societies. Deciding whose oppression is primary or worse (or older, or more unjust, or however else you want to define it) is not all that helpful if your goal is the abolition of class-based societies.
On the other hand, if your goal is to carve out a little more breathing room within the structures that govern (and control, and, yes, oppress) people's lives, then you might want to target a specific area with all the guilt-mongering that your power-hungry clique can manage, all to better your own condition within those structures. That's what identity politicians do. This is the problem with second-wave feminism. This is the problem with affirmative action. This is the problem with the idea of reparations for slavery. This is the problem that leads to the uber-PC discourse around language. All of this relies on white guilt as the prime motivator. All of this relies on lawsuits and legislative solutions.
Checking your privilege -- by acknowledging that your interlocutor is marked by visual inclusion in historically powerful groups -- is great, but not when it's used to shut people up who have unpopular (in this case, anti-liberal post-left anarchist) opinions.
Identity politics in theory could be the self-organization of a self-identified group of people, who've chosen the criteria for inclusion. But identity politics in practice looks an awful lot like lobbying lawmakers to pass ameliorative legislation to protect historically oppressed/marginalized minorities.
