Actually.... the nearer analogy might be how you relate to a team from the same conference representing the conference, as that's something you will have encountered. The chances are you can cope with that provided that the team concerned isn't from your own division?
I think one of the big differences is that a lot of the America rivalries are pretty well ring fenced within the sport. A lot of the British ones go back centuries and have roots way beyond the sport which draw on history, economics, culture, and religion. In fact a lot of them seem crazy all these centuries on, but they're handed down over generations even though the inheritting cohorts can't possibly have any connection to some of the derivations. The Yorkshire/ Lancashire one being the most obvious which traces to 1485 'war of the roses'. Newcastle and Sunderland traces to the civil war circa 1640's. Others like Liverpool and Manchester are Victorian in origin and in this case relate to the building of the Manchester ship canal to by pass Liverpool. Portsmouth and Southampton is a labour dockyard dispute when Southampton workers broke a Portsmouth strike, and as for the Glasgow teams of Rangers and Celtic, lets not go there!!!