> I find it difficult to attribute him to either party.
When your mental model is persistently at odds with the way things work out in reality, you need to reexamine your mental model.
Trump thinks he’s a Republican. A large majority of Republicans enthusiastically agree. I don’t know what you mean by “attribute” but however he got there, Trump is very definitely a Republican.
It might help to recognize that a political party in the US is not one thing. They are a shifting coalition of groups and individuals who have found a way to work together, sometimes comfortably and sometimes not, sometimes for the long term and sometimes not. The policies shift and people change continuously as each tries to get and keep power.