The Chicago Cubs have “Go Cubs Go,” which is unamiguously a team’s theme song. It was written and performed by a fan, it’s specifically about the team, and they play it at home games after every Cubs victory.
Across town, the Chicago Bulls have made the opening notes of the Alan Parsons Projects’ “Eye in the Sky” their hype music when the teams comes out onto the court. It wasn’t written for team, it’s not about the team, and I imagine the guy who wrote it wouldn’t watch a Chicago Bulls game if it was in his back yard and the beer was free. So “EitS” probably blurs the line between “just a song” and “a sports teams theme song,” but nevertheless, that song is forever associated with the Bulls.
Across the Atlantic, a few fans of Wrexham AFC put together a little ditty 'bout their team, and it’s become kind of an ersatz theme song.
So what are some other sports teams that have “theme songs”? Obviously the line between a theme song and not a theme song can be a bit fuzzy (see: the Bulls and their song), so use your best judgment.
Until recently the Broadstreet Bullies AKA Philadelphia Flyers played Kate Smith’s rendition of God Bless America to bring them luck. They dropped her recording because of racist language in other songs she had recorded. I don’t know if they still play some other version. The association between the Flyers and the song was highlighted during the 1974 Stanley Cup finals. Kate Smith sang the song to live before Game 6.
Lots in British football(though none of them theme songs in the sense of being written for the team):
Liverpool have “You’ll never walk alone”
West Ham have “I’m forever blowing bubbles”
Manchester City have “blue moon”
All of them ruthlessly parodied by opposing teams (e.g. “you’ll never walk alone” becomes “you’ll never get a job” mocking the Liverpudlian stereotype of being unemployed petty criminals)
Neither Dirty Water nor Sweet Caroline have very long tenures with the Red Sox. Both really started in 1997 and SC wasn’t made an every game thing until 2002.
Of course, nearly all school teams (high school and college) have an alma mater and/or a fight song. My undergrad school, Villanova, actually had two fight songs (V for Victory and March of the Wildcats).
And meanwhile, Sweet Georgia Brown is probably best-known to most folks as the Harlem Globetrotters song. Though it’s debatable to what extent they count as a “sports team”, as opposed to a sports-themed performance troupe.