I think that the basic intuition relating quantum algebra and quantum physics is something like:
quantum stuff = classical stuff + $\hbar$ (something complicated)
where $\hbar$ is a "small" formal variable. In other words, the point is to consider that the mathematical objects everybody knows are only approximations of more complicated objects. Hence, quantum mathematics has something to do with perturbation theory, because most of the interesting objects in quantum mathematics are perturbations of trivial solutions of some problems/equations. Here, perturbation means that these objects are formal power series in $\hbar$ whose constant term is a trivial solution (eg: 1 :) ) of some equation (eg: the Yang Baxter equation).
Hence, as John pointed out, quantum algebra involves the study of objects for which classical properties (eg: commutativity) are "almost" true (ie: true modulo $\hbar$).