"Quantum groups" aren't actually groups; they're Hopf algebras.
*** Warning: not an expert by any means ***
In a lot of contexts in math it's useful to be able to "deform" objects. What this means, in some sense, is that you can consider the family of such objects as some kind of continuous geometric object, and you can move from one object to another via continuous paths. Since you bring up PSL(2, Z), I'm guessing you know about j-invariants, for instance.
The problem is that certain types of objects, e.g. compact groups, are "rigid" -- the geometric space they form is discrete, so you can't imagine a "path" through them that lets you slightly deform them. However, if you take a group algebra (which has a natural Hopf algebra structure), it can be deformed as a Hopf algebra.
Of course, almost none of the Hopf algebras in a neighborhood of a group algebra will be group algebras themselves, but the idea is to image that they are "group algebras of quantum groups." (In reality, there is no such object as a quantum group "downstairs").