I went through a Nick Hornby reading phase, so Arsenal. Not proud of why, but what are you gonna do? Better than because I had their pajamas or something. That said, I don't have allegiances and rival hatred for BPL teams like I do for US sports -- I'll also root for Everton so long as Howard is there maybe thereafter, and my wife has a connection to Sunderland so I also am partial to them, at least when not managed by a fascist.
All the rest of my European favorite teams are kind of random -- with far less basis than Nick Hornby. In Spain, I'll usually root for Real Madrid to lose, and I find Los Leones kind of compelling -- I respect how well they do given their self-imposed roster restrictions, but also like to ruminate about whether it's racist. Palermo in Italy, but I'll support Juve and the Rossoneri over anyone else among the biggest clubs. I liked Dortmund during the Klopp era, but now don't care much about the Budesliga. The only thing I know about Ligue 1 is that I can often find a game to watch on beIN sports when there's no other games on, but I have trouble imagining I'll care much about French football once Zlatan retires.
Oregon is one of my favorite places to visit in the US, and I liked the Beavers back when they played in the Timbers' current stadium. Plus they really helped move along the European vibe for MLS games in their park, so I guess all other things equal I root for the Timbers in MLS.
Edit: Sorry, didn't really pay attention to the fact that we've got a "pick one" sensibility happening. Arsenal.