Why are Hopf algebras called quantum groups? Why are noncommutative nonassociative Hopf algebras called quantum groups? This seems to be a purely mathematical notion and there is no quantum anywhere in it prima facie.

One way that Hopf algebras come up is as the algebra of (real or complex) functions on a topological group. The multiplication is commutative since it is just pointwise multiplication of functions. However, in non-commutative geometry you want to replace the algebra of functions on a space with a non-commutative algebra, giving a non-commutative Hopf algebra.

This relates to quantum mechanics because there the analog of the classical coordinate functions of position and momentum do not commute. Therefore we think of the algebra of functions on a quantum "space" as being non-commutative.