I think the term "coach" stems from the horse-drawn era where passenger conveyances were "coaches." Stagecoach comes to mind. As the new era of motorized vehicles dawned, the term coach was applied to conveyances like busses, etc. Early motorhomes were converted busses (we had one; a 1950 Flxible) and the name carried over to them as well.
As an aside, our converted bus was Gene Autry's old "band bus." My father-in-law bought it at auction and did the conversion, including replacing the old in-line 6 with a more modern Detroit Diesel 6V53. He was a master woodworker and the interior was reminiscent of a fine old yacht; lots of teak and mahogany. Quite a beautiful piece of work. We sold it years ago, but I think it is still in use.
TJ