Not the whole story indeed, but you have to dive into representation theory somewhat to get more: the Fourier transform is more or less the representation theory of the (abelian) group of the translations of your space, thus the homogeneity requirement. The finite-lattice version[1] (a discretized torus, basically) may serve to hint what’s in stock here.
If you like this topic, I strongly recommend you read the references I attached to my comment.
In uniformly curved 2D hyperbolic spaces, it turns out that there is a higher dimensional non-Abelian Fuchsian translation group defined on a higher genus torus.
[1] https://www-users.cse.umn.edu/~garrett/m/repns/notes_2014-15... (linear algebra required at least to the degree that one is comfortable with the difference between a matrix and an operator and knows what a direct sum is)