As the OP points out, it is natural to study tilings from the viewpoint of symmetries acting on a space. From this perspecitve, finite subgroups of $SO(3)$ lead to tilings of $\mathbb{S}^2$ and the wallpaper groups yield tilings of $\mathbb{R}^2$.

Along the same lines, Chapter 7 of **Farb, Benson, and Dan Margalit. A Primer on Mapping Class Groups (PMS-49). Princeton University Press, 2011**, provides a very nice treatment of the problem for surfaces of genus $g\geq 2$ using the mapping class group.

First, they discuss the $84(g − 1)$ theorem, aka Hurwitz's automorphisms theorem, which shows that the orientation preserving symmetry group of $S_g$ is at most $84(g-1)$. Using (orientable) covers of the orbifold quotient of the $(2,3,7)$ triangle group one can see this bound is sharp.

Also, Chapter 13 of Thurston's notes provides a good bit of background.