We can say that a symmetry group of a square has an operation of rotation by 360° as its identity element. Can we include an operation of rotation by 720° to this group? On one hand, these two rotations are clearly different: we do not want to say that 720° rotation is unneeded because it is a combination of two rotations by 360° (the same reason we do not exclude 360° rotation because it is two 180° rotations). This means that in a sense 720° is also a symmetry. On the other hand both 360° and 720° rotations are identity elements, which is impossible by definition of a group, where an identity element is unique.
Does this mean that having an additional 720° rotation simply makes no sense? What does a group become if it has both 360° and 720° rotations?
The concept of "angle of rotation" is not well-defined. The rotation by $0^\circ$, or by $360^\circ$, or by $720^\circ$, or by $(360 \times n)^\circ$ for any integer $n$, all produce the same function from the square to itself, and that function is the identity function.
The key fact to focus on here is that a symmetry of the square is a certain kind of function whose domain and range is the square. All that matters when deciding whether two functions are the same is that those two functions produce the same output whenever they are given the same input. It is quite possible to have two formulas which appear different, but which nonetheless represent for the same function. For example, if we represent the square as the Cartesian product $[-1,+1] \times [-1,+1]$ in the plane $\mathbb{R} \times \mathbb{R}$, then the identity function $$f(x,y) = (x,y) $$ is the same as the "$720$ degree rotation function" which is given by the formula $$f(x,y) = (\cos(720^\circ) \, x + \sin(720^\circ) \, y, - \sin(720^\circ) \, x + \cos(720^\circ) \, y) $$ This should not be a confusing issue, because this is something you probably encountered long ago. For example, the real valued function $g(x)=2x$ is the same as the function $g(x)=3x-x$, despite the different formulas.