The problem can be rephrased as follows: when you consider a single photon, where does it "know" from whether it can go through a polarizer or no? Is there some vector property of a single photon, that must be aligned with the polarization plane of the filter? What is this property?

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This is incorrect. If you have a collection of photons in which half are left hand circularly polarized ($L$) and half are right ($R$), then you have unpolarized light (not linearly polarized). If you have linear polarized light, then each photon is in a (quantum) superposition of R and L at the same time. It is equally true to say that circularly polarized light is in a superposition of horizontally ($H$) and vertically ($V$) polarized light.

But now my question is - if my above understanding is correct, then how can be linearly polarized light different, than not polarized at all? Both have the same amount of photons having spin parallel and antiparallel to the velocity.

If the light is unpolarized then if you try to measure the polarization in the $L/R$ basis, half the light will be found in each path (just like linear polarization), however if you try to measure $H/V$, half the light will also be measured in each path in this experiment as well (unlike linearly polarized light).

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In other words - what is the relation between spin of each photon (or distribution of spins of all photons) and polarization plane?

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What this means is if you take circular light and shine it on a linear polarizing beamsplitter, half will go in each path (i.e. each photon's wavefunction will collapse to either $H$ or $V$ polarized with 50% probability). However if you tried to measure or seperate the light based on $L/R$, all the light would be measured in one path.