Is the phase shift just the angle of the impedance vector?

``Phase shift'' can actually refer to different things in different contexts. For AC Ohm's law, there is a phase shift between current $\hat{I}$ and voltage $\hat{V}$, and in this case, indeed, the phase shift is the phase angle of the impedance vector $\hat{Z}$.

However ``phase shift'' can also refer to the relative phase of input and output voltages, in which case it's the phase angle of the transfer function $\hat{H}(j \omega)$.