My answer is more concerned with your second question ``Why this should "obviously" be a construction worth looking at, and why it should be useful and meaningful."
I would quote Arnold who, by saying that Mathematics and physics are the two opposite sides of a same medal, conveys the platonistic idea that the coherence and unity of mathematics comes in fact from the coherence and unity of nature, since it is the natural language to describe it.
If one believes in this, one realizes that most of our mathematics is classical, in the sense that most of mathematical objects come from the study of concepts originated in classical mechanics (geometry, Lie groups, ...). But it is known since the early 20th century that classical mechanics is the shadow of quantum mechanics.
Therefore, there should exist a whole brave new world of quantum mathematics, of which classical mathematics should be the ``semiclassical limit".
The paper of Bayern, Flato, Lichnerowicz and Sternheimer gives a paradigm to explore it : they show that quantum mechanics can be interpreted in terms of deformations of associative algebras of classical commutative algebras of observables on Poisson manifolds.
Therefore, an approach to quantize a mathematical concept is to encode its structure in terms of properties of an algebra of functions, and deform this algebra to a non commutative algebra with similar properties. If you apply it to the algebra of functions on a Lie group, you arrive to the concept of quantum group.
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