A polariser causes incoming light to decohere into a classical probabalistic distribution of orientations. The polarisation space of light has two (complex) dimensions. A polariser produces very different outcomes for light which is aligned with the polariser, and light which is aligned orthogonally to it. The interaction of the polariser with the light and with the broader environment causes quantum decoherence, which is the phenomenon where a quantum superposition `collapses' into a classical distribution. (Instead of a coin being in a quantum superposition of Heads and Tails, the universe splits into two parallel universe, where in one it's 100% Heads and in the other it's 100% Tails).
Let's assume monochromatic light with a fixed direction, shining straight into the polariser at an orthogonal angle, and an infinitely thin polariser located at $z=0$ for concreteness. The polarisation of any wavepackets of light at $z=0$ can be decomposed into a linear combination of two vectors (with complex components). To anthropomorphise the situation and make it relatable, a polariser takes the polarisation vector, decomposes it into 2 vectors in a particular basis (one that aligns with the polariser orientation, and the orthogonal one) and then throws out the half that's aligned the wrong way.
So you see, rather than thinking about light having just the right polarisation to pass through, you should be thinking about what the length is of the vector that projects onto the polarisation axis. (Slightly rough picture but it conveys the correct mathematical description, without overcomplicating with complex numbers etc.)
We would call something a 'polariser' when we observe that it causes light to behave as described above. At a more technical level, what's happening is that a polariser is a system which causes decoherence of light into the component that aligns with the polariser direction, and the orthogonal direction. Then, classical probability theory takes over: in rough terms, each photon has a % chance to be aligned correctly, and if so then it passes through.