What Scott's comment is getting at is that you need to have an abstract definition of "quantum Lie group" if you want to have a classification result. As the theory of quantized enveloping algebras and quantized coordinate algebras is currently formulated, this is not really how it works.

Rather, you start with a (finite-dimensional, semisimple, complex) Lie algebra $\mathfrak{g}$, and from its Cartan data you write down the Hopf algebra $U_q(\mathfrak{g})$. I assume here that $q$ is not a root of unity. This is functorial in the somewhat limited sense that if the Dynkin diagram for $\mathfrak{g}_1$ includes into the Dynkin diagram for $\mathfrak{g}_2$ then there is a corresponding map of Hopf algebras $U_q(\mathfrak{g}_1) \to U_q(\mathfrak{g}_2)$ (so diagram automorphisms give rise to Hopf algebra automorphisms). However, this is different from defining a class of Hopf algebras satisfying some properties and then showing that they must all be of the form $U_q(\mathfrak{g})$ for some $\mathfrak{g}$.

If you take $U_q(\mathfrak{g})$ as given, however, then you can construct quantized coordinate algebras corresponding to all of the connected groups $G$ which have Lie algebra $\mathfrak{g}$.

First, let $G$ be the connected, simply connected Lie group with Lie algebra $\mathfrak{g}$. Now $U_q(\mathfrak{g})$ has (morally) the same finite-dimensional representation theory as $U(\mathfrak{g})$ does. If we restrict to Type 1 representations, i.e. those in which the Cartan generators $K_i$ act as powers of $q$ on weight vectors (as opposed to acting as negative powers of $q$, which is well-defined since $q$ isn't a root of unity), then the finite-dimensional representations are in 1-1 correspondence.

For each of these Type 1 representations, we can define matrix coefficients. Say $V$ is a finite-dimensional representation of $U_q(\mathfrak{g})$, and let $v \in V$ and $\phi \in V^*$. Then the associated matrix coefficient is $ c^V_{\phi,v} : U_q(\mathfrak{g}) \to \mathbb{C}, $ defined by
$$ c^V_{\phi,v}(X) = \phi(X\cdot v). $$
These matrix coefficients actually live in the finite dual $U_q(\mathfrak{g})^\circ$, and you define $\mathcal{O}_q(G)$ to be the (Hopf) subalgebra of $U_q(\mathfrak{g})^\circ$ generated by all of the matrix coefficients of all of the finite-dimensional representations. (Of course it suffices just to take irreducibles.) This gives you the quantized analogue of the Peter-Weyl decomposition of the algebra of functions on $G$.

Now there are other connected, but not simply connected, groups which have the same Lie algebra. Let's pick one and call it $G'$. Since $G'$ is a quotient of $G$, then (classically) the coordinate algebra $\mathcal{O}(G')$ is a subalgebra of $\mathcal{O}(G)$. In terms of the Peter-Weyl decomposition, $\mathcal{O}(G')$ is generated by matrix coefficients of all of the finite-dimensional representations of $\mathfrak{g}$ which integrate to give a representation of $G'$. Then you can define $\mathcal{O}_q(G')$ to be the subalgebra of $\mathcal{O}_q(G)$ generated by only those matrix coefficients.

Anyway, I don't know if that's what you were looking for exactly. Hope it's helpful. If you think of a more precise question to ask then you may find more answers forthcoming.