What you're describing is not a symmetry in the sense of Noether's theorem of the (classical i.e. non relativistic and non quantum) system, but it is a symmetry of our description of the system.
A good example is provided when the Hamiltonian can be written as $H=T+V$ where $T$ is a kinetic energy and $V$ is a potential energy. Modifying the Hamiltonian by adding a constant to the potential energy does not change the behavior of the system in either the classical case or in the non relativistic quantum case.
This sort of symmetry is called a "gauge symmetry" and is discussed, in the quantum mechanics case, at length in Sakurai's quantum textbook "Modern Quantum Mechanics":
See section (2.6) page 123, "Potentials and Gauge Transformations", which describes the difference between the classical case, where changing the potential has no physical significance, and the quantum case, where changing the potential changes the phase, but doesn't result in physically observable consequences.