From the Super Mario Wiki
The Backscratcher is an item in the SNES and PC releases of Mario's Time Machine. It features three long teeth at the end of the wooden stick. It is given by the owner of a music store in Vienna. He hands it out as an attempt to attract customers through novelty items such as the Backscratcher and the Fan. The only person in the area who wants the item is the owner of the Swan Inn, who mentioned that he was "itching" to talk about Ludwig van Beethoven but refused to speak further. Once Mario presents the Backscratcher to the hotel owner, he appreciates Mario's play-on-words (a backscratcher being used to scratch itches), and gives Beethoven's Monocle to Mario due to the latter's interest in the composer.