Ah, the difference between the public and private spheres, and the difference between collectivism and individualism. I assume that Laurie Penny is saying that all politics are identity politics because the political is personal. Politics is about real people, who fit into a variety of different identities. I also assume that she approaches politics from a socialist, collectivist angle, focusing on groups, rather than subscribing to an individualist philosophy.
I think you may be correct. I also think she needs to chill out.
Perhaps she needs a G&T? I make excellent G&Ts.