I heard a song this morning by a band called The National.
Not the Nationals, but close enough to make me think of the baseball team from DC.
The Eagles are a football team and a rock band.
What else?
(Could probably have gone either here or Cafe Society - might have gotten different responses there too!)
Are we limited to SUCCESSFUl bands? If not, Adrian Belew of King Crimson has a band on the side called the BEARS.
Can we count groups where a team name is part of the band’s name? If so, there’s Paul Revere and the RAIDERS.
Back in the 80s, there was a hard rock band caled GIANT. Does that count?
Chicago.
The Penguins were a popular doo wop group of the Fifties, and are now an NHL hockey team.
We can count whatever we want to count. 
Well there is/was a band called The Stampeders (Sweet City Woman). They were from Calgary, home of CFL team Stampeders.
A quick Google shows that every team in the NHL’s Atlantic Division (Devils, Rangers, Islanders, Penguins, Flyers) shares a name with a band, though it might depend on the as-of-yet undefined acceptable definition of band-- for instance The Rangers were a “Czech pop folk” group whose time was the 70s and early 80s and The Flyers are an indie band out of Texas who, for all I could find, don’t exist outside of a MySpace page.
some that don’t quite fit your definition
John “Cougar” Mellencamp
“Boston” Red Sox
Maryland Terrapins - The Turtles
New York ‘‘Flyers’’ - Philadelphia Flyers
How about sports teams that were named after a song? I give you the St. Louis Blues.
There is a British electropop band named St. Etienne, which translates from the French as “St. Stephen” but in this case was named after the French soccer team of that name.
Night RANGER?
And right after splitting with CCR, John Fogarty had a short-lived band called the Blue Ridge RANGERS. (It’s unclear, according to some reference books, whether there really was a band or whether the Rangers were really just John Fogarty and a few session men.)
Or named after a genre, like the Utah Jazz.
Ryan Adams & The Cardinals
Bennie & The Jets?
And coincidentally, the Eagles (band) played the first event at Invesco Field, the new home of the Denver Broncos.
My high school mascot was SCORPIONS
They might be Giants.
bob marley and the whalers…
Well…They Might Be Giants. 
Gah, scooped!
There’s The Commodores and the Vanderbilt Commodores.
Johnny and the Hurricanes (Carolina Hurricanes).