We have neither absolute knowledge, nor logically necessary arguments to any such effect. Empirically we are pretty confident of a high degree of isotropy, and many cosmological theories sit comfortably with isotropy, but there are some observations and some theories suggesting various degrees of anisotropy.

As for fractal structures, why do you suggest that they are fundamentally incompatible with isotropy? Remember that our ideas on the isotropy of the universe do not assert detailed isotropy, but large scale isotropy. We know very well that on a fine scale the universe is anisotropic. If you don't believe me, step outside, day or night, and see whether there is any direction in which you see an isptropic universe. There is nothing of the type. On a similarly coarse scale there is no reason to deny a fractal geometry of the universe. In fact there is pretty good evidence for fractal structure on a galactic and super-galactic scale.

I am not sure what exactly you have in mind.