Polarization is not more "classical" than frequency or wavelength is. These characteristics are not fundamentally impacted when quantizing. Every EM wave can be decomposed in two waves, one that has left-hand circular polarization and one that has a righ-hand circular polarization. When you quantize a classical wave, you quantize the amplitude of the wave, but you don't touch the polarization (see for example https://en.wikipedia.org/wiki/Quantization_of_the_electromagnetic_field ), just like you don't touch frequency or wavelength.
It turns out that polarization can be associated with the spin of the photon, however there is a one to one relationship between polarization states and spin states, so basically they are both describing the same physical phenomenon.