The description of "Short Rest", on page 186 of the PH, contains the following precise verbage:
A character can spend one or more Hit Dice at the end of a short rest, up to ...
Because of the identical phrasing, the rules as they are written allow for a bard's Song of Rest ability to trigger only for some other gain of HP with the same phrasing. If a fighter is all out of Hit Dice to roll, but somehow still has their second wind, a potion of healing, or an ally with a cure wounds spell, they wouldn't gain the bonus HP from the bard's song.
There may be spells or magical items with the same phrasing as the bard's ability and the standard short-rest HP gain, but none of the ones I checked in the PH would qualify. (Except for maybe an oddly worded wish.)
It would not be game-breaking to allow this as a house rule, though. (Nor would it surprise me to find it changed via errata.)