I’m not talking about a team known as “The Eagles” with a guy in an eagle suit. I mean they also have a living, breathing eagle, too. Or maybe they have an unofficial/official team “pet” that has no bearing on or relationship to the team’s nickname (obviously Wrexham aren’t going to have a real dragon).
St. Louis City SC has Petey, a Labrador retriever who is the Official Team Pet (and also a metaphor for a sponsor and a big business in St. Louis: Purina).
Any others?
My high school had Panthers.
My daughter’s high school had Tigers.
The Brewers, up until a few years ago, had Hank.
Since 1956, the University of Georgia Bulldogs have had a string of live bulldogs, all named Uga. They’re up to Uga XI.
My high school, for a few years, had a succession of Bengal tiger cubs, until folks finally realized what a spectacularly bad idea that was. Sure, tiger cubs are cute, but they don’t stay cubs.
Similarly, the University of Texas Longhorns have had a string of live longhorn cattle, all named Bevo, dating back to 1916. They’re up to Bevo XV.
Tusk.
Live Razorback mascot.
(U of A)
The University of Southern California has a white horse ridden on by a Trojan.
The Seattle Seahawks have a hawk that flies around before each home game.
The University of North Alabama had live lions until quite recently, and is looking at bringing them back.
Pretty sure both Army and Navy have live mascots, a mule and a goat.
The University of Wisconsin Badgers briefly had a live badger mascot at football games, but as American badgers are aggressive, this turned out to be a bad idea.
Since 1949, the mascot has been a student in a “Bucky Badger” costume.
The University of Colorado has a buffalo (bison) as a mascot. It’s always named Ralphie, and the current one is Ralphie VI. The animal storms the field before each football game, with several students holding a ‘leash’.
video here https://www.youtube.com/watch?v=n_881gHgSHU
Perhaps a better-known bulldog mascot is Handsome Dan. Since 1889, successive bulldogs have been Yale’s mascot. They’re now up to Handsome Dan XIX.
The Texas A&M Aggies have had a long line of collies named Reveille as their mascot.
Interestingly, it is Aggie tradition that if Reveille enters a classroom, and barks, then the professor has to cancel the class session and dismiss the students at once.
But did they have actual panthers and tigers on the sideline at games? Because that would be cool! 
LSU has had a series of live Tigers named Mike, housed in a facility on campus and brought out to both home and away games, since 1937. They are currently up to Mike VII, who has been on campus since 2017.
Mules.
University of Southern Arkansas Muleriders.
Barns full of them. Miss Mulerider rides one at home games. Apparently it’s famous for being real mule-headed at games, the spectators love it.
My hometown Texas A&M Kingsville (Texas A&I back in the day) Javelinas have a javelina that attends all the football games. Porky, as the mascot is known, is kept in an enclosure attached to a golf cart. Whenever the Javelinas score a touchdown Porky makes a lap around the field while the person driving the golf cart rings a bell.
The Miami Dolphins during their early years had a live dolphin in a tank in one end zone, named “Flipper” natch after the TV character (on load from the nearby Seaquarium).