My great-grandfather was recorded as "Professor of Music" on the 1861 census. I'm quite sure he wouldn't have trained anywhere, because he would not have had the money to do so. He learned to play the oboe while working in a woollen mill, became so good at it that he gave lessons to others, and eventually found that this became a full-time career. He also played the piano and organ, and later opened a shop selling, tuning and repairing musical instruments, played the organ for a local church, founded and conducted an orchestra and directed a choir.
Another very distant relative was of Italian birth and was described as a "Street Musician" in one census and and "Ice Cream Seller" in the next! His home had a number of boarders described as "Musicians", probably hurdy-gurdy men, rather than serious musicians like my great-grandfather.
I think the interpretation of "Professor" as a teacher is probably nearer the mark.