In this article, Appiah suggests that the claims of representation that underlie identity politics (“I as a so-and-so say that…”) imply something like “as an (x) am in a position to speak for random person (a), who is also an (x)”. In other words, political representation means ‘speaking for’. On this view, e.g., if I claim to speak as a heterosexual man when I offer some witticism or piece of prosaic advice (“As a man, I don’t care about gender-labeling washrooms”), then I am speaking for men. Meaning, I guess, that I’m saying the sort of thing that other men would also say.
I’m sure some people do talk and reason in that way, but I also think it’s just one way of speaking among others. So, e.g., I think of these claims as usually about how I and (a) both have equal though partial authorship rights over the experiential meaning of (x), as opposed to person (b) who is not an (x). It involves speaking for your role, not necessarily speaking for others who also have that role. So, to use the same example as above: sometimes, if I say, “As a man, I don’t care about gender-labeling washrooms”, I am not speaking for men, but speaking in my role as a man, which may or may not generalize.