Question about the wave nature of light

You are right that the oscillations of the electromagnetic field need not have any spatial extent. The oscillations, as you point out, are in the strength of the electric and magnetic fields. If I understand your question correctly, you are asking why then can some objects distinguish between the two different polarizations of light.

This is because anisotropic materials (those which do not have exactly the same structure in all directions) can act differently to electric fields which lie along their different axes. In the image below the material in the middle does not support an electric field in the horizontal direction. When you try to establish an electric field in the horizontal direction, energy is taken out of the source which is trying to establish it (the light in this case). So, this material, which acts as a typical polarizer, only allows light which is polarized in the vertical direction to pass.

There are other ways to distinguish between the two different polarizations of light. Birefringent crystals, for instance, have a different index of refraction for the two different polarizations of light. They therefore deflect the two different polarizations of light by a different amount.