Is there any way to improve upon the fit if we know that e.g. y is n times as noisy as x? Or more generally, if we know the (approximate) noise distribution for each free variable?

Yeah, you can generally "whiten" the problem by scaling it in each axis until the variance is the same in each dimension. What you describe is if x and y have a covariance matrix of like

    [ σ², 0;
      0,  (nσ)² ]

[1] https://en.wikipedia.org/wiki/Whitening_transformation

> Or more generally, if we know the (approximate) noise distribution for each free variable?

This was a thing 30 odd years ago in radiometric spectrometry surveying.

The X var was time slot, a sequence of (say) one second observation accumulation windows, the Yn vars were 256 (or 512, etc) sections of the observable ground gamma ray spectrum (many low energy counts from the ground, Uranium, Thorium, Potassium, and associated breakdown daughter products; some high energy counts from the infinite cosmic background that made it through the radiation belts and atmosphere to near surface altitudes)

There was a primary NASVD (Noise Adjusted SVD) algorithm (Simple var adjustment based on expected gamma event distributions by energy levels) and a number of tweaks and variations based on how much other knowledge seemed relevant (broad area geology and radon expression by time of day, etc)

See, eg: Improved NASVD smoothing of airborne gamma-ray spectra Minty / McFadden (1998) - https://connectsci.au/eg/article-abstract/29/4/516/80344/Imp...