Axiomatic definition of quantum groups

I would have liked to write this as a comment, but with my points tally I can not. So writing this as an answer.

In quantum groups, we are probably at a stage group theory was, say in the first half of the 19th century (see here and here), where we have several important classes of objects that we more or less agree should qualify to be called quantum groups, but it is not clear yet if we are anywhere close to having a single set of axioms that will cover all these classes. It is in fact far from clear whether the union of all these `subclasses' will be part of one single meaningful class.

I would say that if you are looking for a concrete definition then it's better to adopt the Tannakian point of view and to focus on the category of representations of the quantum group rather than on the algebra itself. So take as your fundamental object a tensor category (a special type of rigid abelian monoidal category - see here for details). A "quantum group" is then some way of realising the category as a category of representations or corepresentations. There can be different algebras which do this job, and they can come in different flavours, such as Hopf algebras or compact quantum groups in the sense of Woronowicz. This allows one to view the various quantum groups floating around as tools to study the category itself, removing the need for any axiomatic definition.

If the object is really deserving of the name quantum group, then the tensor category should be braided, as is the case for quasi-triangular Hopf algebras and their category of modules. (see the comment of Sam Hopkins above.)

A cheap, soft, quick but objectionable meta-definition:

An algebra of functions on a quantum group is an algebra that satisfies some specific axioms such that whenever an algebra satisfying those same axioms is commutative, it is an algebra of functions on a group (and e.g. the group multiplication given as the transpose of the comultiplication); and whenever two commutative algebras satisfy those axioms are isomorphic as objects satisfying those axioms, their underlying groups are isomorphic.

Not satisfactory but a start.