When studying the dihedral group of a square, do we consider only vertices or the whole points which the square covers? Because the vertices of square also gives the same symmetries.
When studying the symmetry groups of a whole geometric object, the symmetry of all points of that object is considered.
Take e.g. a square that contains a non-symmetric pattern on its surface: it has a different symmetry group than a square: its symmetry group is the identity group.