Generally, harmonic signals have a phase: the derivative of phase over time is the frequency of such a signal, I.e. a sine or cosine.
Now, not all periodic signals are harmonic signals. But we can decompose any periodic signal into a sum of harmonic signals (that's the Fourier series of the periodic signal).
It's easy to see that every harmonic component of that sum can have its own phase. There isn't such a thing as "the phase" for a non-harmonic signal.
However, it can be useful to think in terms of the phase of the fundamental frequency, I.e. the harmonic component of the same frequency as your periodic signal, and call that the phase of the non-harmonic periodic signal. However, that fundamental can, when the periodic signal is composed of two periodic signals whose periods are rationally related but not multiples of each other, be of zero amplitude and then you can't really define the phase of it.
So: long story short: strictly speaking, only harmonic signals have phase. Some periodic non-harmonic signals can be assigned a property analog to that concept, but not all of them.