What about the Fuchsian groups make them stand out? Why do we stop at Fuchsian groups (I.e. discrete subgroups of automorphisms of the hyperbolic plane) when we study things like quotients and what not?

Is there a maximalist or universality property behind that distinction?
Who stops? Not me. We continue on to:


Lattices in Lie groups 
Cayley graphs
Geometric group theory


and on and on from there...
Depends on what kind of whatnot you want to study. It's a fact that any hyperbolic Riemann surface (that is, any connected complex manifold of dimension one except the Riemann sphere, complex tori, or the cylinder) can be written as the quotient of the upper half-plane by a Fuchsian group. 

So the study of Riemann surfaces is basically equivalent to the study of Fuchsian groups.