His name, both of his jobs, and his creepy obsession with surgery are likely a reference to medical practices in colonial times, which appears to be roughly around the time the show takes place in. Regular doctors considered themselves to be above surgery, so barbers were instead called upon when an operation needed to be done since they were skilled with sharp tools and making precise cuts.
He is accompanied by three other singing barber's assistants in the first episode he appears in, making him part of a quartet, another recurring gag among old timey barbers. The three singers are strangely missing from all of the other episodes that he appears in afterwards.
Doctor Barber, as seen as a background character in Punch Time Explosion.