Nope, `fmap` is indeed short for functor map. It's a historical accident in Haskell as `map` was specialised for lists (ie. `[]`) from the start and later, when mapping would be generalised to all things Functorial, it needed a new name so as to not break everything, hence `fmap`.
No, "fmap" is actually the name in Haskell given to the function defined by the Functor typeclass. It was named "fmap" ("f"unctor map) because "map" was already taken by the map specialized to lists.