With your example I think it would be fine to have a general identity and promote the opposite site since most people would consider themselves more of a dog or cat lover and having your identity match wouldn't be as relatable when promoting an opposite site. Its not trustworthy for a cat lover to push a dog site, you know?

Otherwise it may or may not work. If the two are connected but slightly different a more specific identity might work better, but having an easy to identify identity across many sites could be better.