Yes, every object has to be keyed by name in the animation clip, which causes problems with re-using a clip on another character, particularly with facial hair.
That's what the
Animation Retargeter is for, it will pull the blend shape tracks from the chosen clip and generate tracks for any matching blend shapes in other objects, for the current character in the importer window, as well as some other corrections:
Blend Shape RetargetingOlder video, but still relevant: