OK, so to check that I understand this:
Imagine two cameras, one a 32MP APS-C (e.g., the M6-II or 90D), and the other a 32MP full frame (not sure there is such a beast), the sensors are the same "generation" and so on. Stick a 500mm lens on the crop camera, site yourself somewhere, and take a picture. Then grab the full frame, put an 800mm lens on it, and take a picture. Assume both are at the same f/ ratio. They should look identical in composition and perspective.
They wouldn't. They have different entrance pupil sizes, so they will render the infinity differently.
I would expect, in this case for the crop picture to look a bit noisier, right? Both pictures have the same number of pixels, but in the FF case they're "bigger" pixels.
It will be noisier because it has the same angle of view, but smaller entrance pupil, so it gathers less amount of photons through its entrance pupil. It does not really matter on sensor of what size these photons are projected.
If, instead of the same f/ ratio, you specify the same entrance pupil size, you will get exactly the same picture and exactly the same noise, as long as your smaller sensor can cope with the increased exposure (and decreased ISO to match it without losing photons in ND filters).
A smaller sensor needs a higher light flux density to receive the same light flux, but
as long as the lens can project it and the sensor is not overflown by it (both
limitations benefit the larger sensor area), the smaller sensor will render the same image with the same noise.