Former professional wrestler Jesse Ventura, who voices The Finger, used the moniker "The Body" during his wrestling career. When The Finger is fighting Michelangelo and Donatello in the woods, he uses some wrestling moves on them.
In the scene when Donnie and Mikey fight The Finger they use the old Adam West Batman "POW"/"BANG" imagery, I'm assuming as an homage to Bill Finger (co-creator of Batman)
The Finger is somewhat of a parody of Kraven The Hunter, a supervillain of Spider-Man. His appearance resembles Zangief, a Wrestler from the Street Fighter franchise.