Is there an analogy of quantum mechanical phase symmetry in classical physics? In quantum mechanics, phase symmetry is the fact that the state of a system can be multiplied by a length 1 complex number without changing the probabilities of observables.
Is there an analogous symmetry concept in classical physics?
 A: A classical analogue to unitary transformations, which uses the principle of phase indifference with some restrictions (for unitary operators  $\hat{U}^+\hat{U}$ must be equal to $1$), would be orthogonal transformations of real vectors. There's a good text about this in the book Student Friendly Quantum Field Theory by Robert D. Klauber.

A unitary transformation is called unitary because its operation on (transformation of) a state vector leaves
  the magnitude of the state vector unchanged, i.e., the state vector magnitude is multiplied by unity. It is the
  complex space analogue of an orthogonal transformation in Cartesian coordinate space, which, when acting on
  a (real number) vector in that space, rotates the vector but does not stretch or compact it. A unitary
  transformation can be thought of as "rotating" a (complex number) state vector in Hilbert space (the complex
  space where each coordinate axis is an eigenvector) without changing the "length" (magnitude) of the vector.

A: For any wave-like object, one can change the phase without changing the intensity. This holds for instance for for electromagnetic waves, acoustic waves & water waves.
