Why is there a non-zero phase error in a second-order PLL?
A frequency step is identical with a ramping phase (phase is the time integral over frequency). Therefore, speaking about phase errror (for a frequency step) we do NOT speak about a "static" but about the "dynamic" error (or phase "tracking" error).

Applying the "Final Value Theorem", we find that (a) for a simple lag-filter we have a finite (tracking) phase error, but for a PI filter the dynamic phase error is zero.  

Explanation (Loop gain): LG(s)=H(PI)*H(VCO)=K1(1+1/sT1)*K2/s.

Because of Delta(phi)=Delta(w)/s the "s" in the denominator of LG(s) cancels and the "Final Value Theorem" for the phase-transfer function (with s=0) gives a finite (correction: infinite !)value for the loop gain LG in case of a PI controller).