The definition of terms "classical" and "quantum" is not always clear-cut. Of course one might say that everything is quantum and nothing is classical, but I think the distinction is worth making; it draws our attention to those parts of quantum theory that are well-captured by classical concepts.
The notion of a continuous field, characterised by a small set of real numbers at each point, and able to be observed without disturbing it, is "classical". Classical electromagnetism (Maxwell's equations; Lorentz force equation) is, then, classical physics.
If we take it that a linear polarizer transmits just the part of the electric field in a wave which is aligned in a given direction, then the $\cos(\theta)$ law for the amplitude, and therefore $\cos^2(\theta)$ for the intensity, follows immediately. In this sense it is classical.
If, on the other hand, we look into the physical interaction in the polarizer that causes it to have this kind of transmission, then in some cases we would struggle to come up with a good model employing only classical concepts.
In the case of radio waves and a polarizer just consisting of a set of parallel conducting wires, a classical picture is quite good. In the case of light waves in a birefringent material you can also use classical models of the fields, but you have to take the birefrigence itself as a given. In other cases the light-matter interaction is more 'quantum'. The inverted-commas are there to remind that the issue of specifying whether or not some process is classical is not a