They're technically the same empire but split up, so yeah. Think of each as mini-factions of one larger faction. Each faction's ruler is claiming to be the true Emperor. In Warband terms they would each be claimants of each other's thrones, and therefore share a culture.
I suppose it could be more interesting for each to have a slightly different flavor, like one of them being more focused on cavalry or something, but I don't personally think it's necessary. I also personally feel that uniting the Empire (which is a large part of the story btw) would become a goal for many players, and having them all be uniform makes that much easier to manage.