Which topics engage people's identity depends on the people, not the topic. The most intriguing thing about this theory, if it's right, is that it explains not merely which kinds of discussions to avoid, but how to have better ideas. If people can't think clearly about anything that has become part of their identity, then all other things being equal, the best plan is to let as few things into your identity as possible.
Do religion and politics have something in common that explains this similarity? One possible explanation is that they deal with questions that have no definite answers, so there's no back pressure on people's opinions. But this isn't true. There are certainly some political questions that have definite answers, like how much a new government policy will cost.