There are two kinds of symmetries, global symmetries and local symmetries. In case of a global symmetry the same transformation is applied everywhere, for example moving all particles of a system to the right by one meter. In case of a local symmetry the transformation can vary over space and time, so you could move different particles by different distances, for example double the x coordinate.

Classical mechanics is invariant under global translations, it does not matter where the system is, here or one meter to the left, what matters are just differences between positions, the distances between particles. On the other hand classical mechanics is not invariant under local translations because it will alter the distances between particles. Local symmetries are a much stronger constraints than global symmetries which are essentially just a special case of local symmetries where the transformation parameters are fixed across space and time.

The wave function has a global symmetry, you can rotate the phase, but phase differences are important and you can therefore not apply different phase rotations to different parts of the wave function. On the other hand the relevant symmetries in gauge theories are local symmetries, so the global symmetry of the wave function is not what makes a theory of quantum physics a gauge theory. In case of electrodynamics it is for example the electromagnetic four-potential that has a gauge freedom, i.e. there are many different electromagnetic four-potentials that give rise to identical physically observable magnetic and electric fields.