Robert, 1) It is not limited to romantic situations. For instance I may hit it off with a new business colleague who shares my passions and attitudes about the job. I think of this more as an expression of how easy it is to make productive and delightful conversation. We think in the same level of abstraction and perhaps have similar interests, goals, and/or methods. Of course if you hit it off in the gym instead of at the workplace then it may be because you find training abilities, methods, et cetera in common and consequently find it enjoyable and productive to train together. 2) If you feel that you have hit it off with someone then you think that there was mutual appreciation. It is possible to take a fancy to someone without them knowing who you are.

"hit it off with someone" is actually seldom used for romantic attraction. It is most of the time about people getting along as friends, workmates and so on, owing to similar personality and disposition. The MerriamWebster dictionary describes it as " to get along well; become friends". And no dictionaries I checked mention anything related to romance. It is also typically a spontaneous and outward reaction (enthusiastic smiles, energetic interchanges, loving to do things together, etc), so I would say it's really different from "taking a fancy/liking to somebody", which indicates something happening more gradually and subtly.

It can be used non-romantically to mean to quickly become friends with someone.