Given the way Premier League works, it's not really geographic any more. "Arsenal" is owned by an American, plays in a stadium named after the airline of the UAE, recruits its team from all over the world, and is named after the Royal arsenal that no longer exists.
Names and history are important. That's why there has been such a fight over renaming US teams with racist names.
I don't think it's a simple as that, and you have to separate the ownership of the club and the makeup of the team from who makes up the bulk of the fans.
The fans don't particularly care that it's owned by an american and sponsored by a UAE airline, because what matters is the coming together of the fans behind a common cause.
The fans do typically come from the area around the stadium in North London.
When you get just a little bit further down the league it's still very much a geographical thing because the teams are less well known and less successful, so are less likely to attract people who want to support "the best team". For example, Southampton FC fans are pretty much exclusively from Southampton.
Further to that, the fans care deeply if the owner/manager/players make efforts to become part of the fabric of the area, otherwise they won't care about them to any great degree (even if a player is world-class and delivers success, without integrating, they'll still be seen as mercenary). Fans, very much the same regardless of level: support is an identity very much tied to a specific place, and a tourist can't really fully participate in that.
Names and history are important. That's why there has been such a fight over renaming US teams with racist names.