It is usually called a gauge symmetry but it is kind of a misnomer and should better be called a gauge redundancy. It is artifical and only due to the mathematical formalism we are using. As an analogy, probably not a really good one, is that you could decide to describe temperatures with a complex number and just say temperature is invariant under translation along the imaginary axis. Sure, the symmetry is in your mathematics but it doesn't really tell you anything new or interesting. Besides of course that the imaginary part is actually redundant and you better described temperature with a real number instead.
That would not work in the same way as a typical gauge theory though - typically the symmetry of a gauge is for shifts of the entire system along the gauge - changes along the gauge locally do have physical meaning. For example, in an electrical diagram you can arbitrarily shift every voltage in the system by a billion Volts (ground rail is 1b Volts, 5V rail is 1b+5V, etc) and the system will behave identically. The gauge quantity is real and not redundant, it’s just that it doesn’t have a known meaningful absolute reference point.
Gauge symmetries are local symmetries. In case of the complex temperature field you can change the imaginary part of the temperature at each point independently without any physical effect. Global symmetries on the other hand are meaningful, for example spatial translation and rotations and time translations in Newtonian spacetime are associated with the isotropy and homogenity of space and time and also the related conservation of momentum, angular momentum, and energy. And of course your example of changinging the reference potential is also a global symmetry, not a gauge symmetry.