Yes, for a *classical* field (*e.g.* a large photon number, coherent state), in principle the phase can be measured in the sense that one can in principle observe that the electric (magnetic) field at a given point oscillates sinusoidally with time and one can measure at what times the field is zero and when it reaches its maximums. 

If you don't believe this for light, simply imagine the experiment with microwaves, or radiowaves. Phase locked loops have been synchronizing oscillator circuits with incoming radio waves for the [best part of a hundred years](https://en.wikipedia.org/wiki/Phase-locked_loop#History). 

So at optical frequencies, the difference is one of technology only, not physics. Optical phase locked loops now exist and realize coherent modulation schemes over optical fiber communication links.

For a quantum state, of course, the state is a ray in projective Hilbert space and is invariant with respect to multiplication by a global phase. No meaning can therefore be given to the absolute phase of a quantum state. Of course, phase differences between the complex amplitudes of base states in superposition are meaningful, and these determine interference effects and the evolution of the quantum state.