That's exactly the situation - my surname, FLESHER, is very often 'assumed' to be FLETCHER - (this is also experienced by many of my relatives).
You are correct FLETCHER is an English surname; FLESHER (originally FLEISCHER) is German .
Footnote: my third GGF, Balser M. FLESHER, was one of the original settlers of Muskingum County, Ohio - in the History of Muskingum County, you will find many references to Balser FLETCHER (and none to Flesher).