Doesn't this imply that equivalent scientific theories may have quite different implications wrt. what abstract objects exist, depending on how exactly they are formulated (i.e. the extent to which they rely on quantifying over variables)?

I'm not sure what you mean by "equivalent" here. If you mean "makes the same predictions" then yes-- but that isn't really an interesting fact. There are empirically equivalent theories that quantify over different concrete objects too. Usually we can and do adjudicate between empirically equivalent theories using additional criteria: generality, parsimony, ease of calculation etc.