Overall, it feels a bi ugly because of the duplicated code

Looks fine to me.

and it will print the warning for every line of an array that has a deficient rank, which can be a lot...

That could be annoying as a user. This happens in results = da.apply_along_axis(nputils._nanpolyfit_1d, ...), right? Can you use a warnings.simplefilter("once") so the error is only issued once - something along the lines of:

with warnings.catch_warnings():
    warnings.simplefilter("once", np.RankWarning)
    results = da.apply_along_axis(
        nputils._nanpolyfit_1d,
        ...
    )

I think this also warrants tests and a whats-new entry.

So while adding tests, I realized there were more bugs concerning deficient rank matrices and full=True output. I believe I fixed all I could find.
However, I was not able to reduce the number of warnings for the case using dask : the computation occurs outside of the catch_warnings() context...
Also, there is a bug in the dask.array.linalg.lstsq output (see issue dask/dask#6516), so if full=True we must use skipna=True with dask. Same when rank != order (deficient rank), dask.array.linalg.lstsq will fail, so we work around by using our (slower) nan-skipping method.

Pity that the dask warnings are not caught but I guess there's not much we can do about that.

Does this now work for the originally posted issue? Could you add it as a test?

@mathause Fixed the tests to catch the warning and added a case for the specific of the issue : too many nans causing a rank deficiency.

thanks @aulemahal!