I think we're sticking on our interpretations of properties A, B, and C. I and the above poster are using them to refer to views fundamental to the political goals of the identity X. So for example saying "if you don't vote for Biden you're not really Black" is saying if you identify as Black, you must have the property of supporting Biden as presumably he is the best candidate for the Black community.
You appear to be using property to refer to other identities. So someone who identifies as both Black and a woman would thus be a black person with with the property of woman. Of course proponents of identity politics want as many people of their identity on their side as possible, regardless of what else they may identify as, but you have to ask what being on their side means. Just saying X should have it better hides an implicit question: what does it mean for X to have it better? Many proponents of identity politics simply assume that everyone of identity X will agree on what they collectively want, but this is to assume they share the same properties A, B, and C which make them want these things.
You appear to be using property to refer to other identities. So someone who identifies as both Black and a woman would thus be a black person with with the property of woman. Of course proponents of identity politics want as many people of their identity on their side as possible, regardless of what else they may identify as, but you have to ask what being on their side means. Just saying X should have it better hides an implicit question: what does it mean for X to have it better? Many proponents of identity politics simply assume that everyone of identity X will agree on what they collectively want, but this is to assume they share the same properties A, B, and C which make them want these things.