This is not really an answer. It's more like a stub.
I hope that by making it wiki, I'll encourage others to contribute.
So there are plenty of things classified by A-D-E (or variants thereof, such as A-B-C-D-E-F-G, or A-D-E-T, or A-Deven-Eeven). In particular, there are plently of things called "the E8 ...".
Instead of making a complete graph, and showing that for any X and Y, "the E8 X" is directly related to "the E8 Y", one should maybe be less ambitious, and only construct a connected graph.
So, our task is to connect "the E8 Lie group" to "the E8 quantum subgroup of SU(2)".
I suggest the following chain:
E8 Lie group --- E8 Lie algebra --- E8 surface singularity --- E8 subgroup of SU(2) --- E8 quantum subgroup of SU(2).
(1) E8 Lie group --- E8 Lie algebra
No comment.
(2) E8 Lie algebra --- E8 surface singularity
Look at the nilpotent cone $C$ inside $\mathfrak g_{E_8}$. That's a singular algebraic variety with a singular stratum in codimension 2. The transverse geometry of that singular stratum yields a surface singularity.
(3) E8 surface singularity --- E8 subgroup of SU(2)
Given a finite subgroup $\Gamma\subset SU(2)$, the surface singularity is $X:=\mathbb C^1/\Gamma$.
Conversely, $\Gamma$ is the fundamental group of $X\setminus \{0\}$.
(4) E8 subgroup of SU(2) --- E8 quantum subgroup of SU(2)
Is this quantization?!?
So let's recall what one really means when one talks about the "E8 quantum subgroup of SU(2)". We start with the fusion category Rep(SU(2))28, which one can realize either using quantum groups, or loop groups, or vertex algebras. That category is a truncated version of Rep(SU(2)): whereas Rep(SU(2)) has infinitely many simple objects, Rep(SU(2))28 has only finitely many, 29 to be precise.
Now this is what the "E8 quantum subgroup of SU(2)" really is: it's a module category for Rep(SU(2))28. In other words, it's category M equipped with a functor Rep(SU(2))28 × M → M, etc. etc.
That's where one sees that "quantum subgroup of SU(2)" is really a big abuse of language.
...so I don't know how to relate subgroups of SU(2) with the corresponding "quantum subgroups".
Can anybody help?
Let $\mathcal{C}$ be the category of finite-dimensional representations of a semisimple Lie algebra $\mathfrak{g}$ and $\mathcal{C}[\![\hbar]\!]$ the ribbon category you mention (which depends on the choice of a Drinfeld associator).
Given an irreducible $\mathfrak{g}$-representation $V$, the corresponding knot invariant obtained from $\mathcal{C}[\![\hbar]\!]$ is the Kontsevich integral evaluated using the weight system coming from $\mathfrak{g}$, $\Omega$ and $V$; this is Theorem 10 in Lê--Murakami's ``The universal Vassiliev-Kontsevich invariant for framed oriented links'' (https://arxiv.org/abs/hep-th/9401016).
The $n$-th Taylor coefficient (with respect to the $\hbar$ expansion) of the Kontsevich integral is a finite type (Vassiliev) invariant of degree less than $n + 1$. But the first nontrivial finite type invariant of oriented knots has degree 2 and so requires working at least with $\mathcal{C}[\hbar]/\hbar^3$.
So, you could consider oriented knot invariants working modulo $\hbar^2$, but they are all independent of the knot.
Best Answer
Without directly answering the question about Drinfeld's paper, I'd emphasize the variations in the definition of "quantum group" in the literature and the resulting variation in discussions about centers, representations, etc. The early work by Drinfeld and Jimbo was partly motivated by mathematical physics, so the $q$ in the definition might be a complex number (say related to the Planck constant). But in later work by Lusztig and others the $q$ is at first an indeterminate and can then be specialized, so that $q=1$ recovers something close to the universal enveloping algebra of a complex semisimple Lie algebra whereas $q \neq 1$ a root of unity leads elsewhere. Jantzen's book mostly follows Lusztig's lead in the treatment of quantized universal enveloping algebras, where a version of Kostant's $\mathbb{Z}$-form leads to finite dimensional versions, etc. (Other generalizations involve affine Lie algebras, while very different "quantum groups" arise from function algebras of algebraic groups.)
In all of these settings it is a problem to describe the center of the quantum group and relate it to the (initially finite dimensional) representation theory. As far as I know, only partial information about centers has been developed so far, while at least for Lusztig's quantum groups the finite dimensional representations have mostly been studied without full knowledge of the center. That has certainly been true in modular representation theory as well.
The good classical prototype occurs in the older work of Chevalley and Harish-Chandra on the usual universal enveloping algebra of a complex semisimple Lie algebra (say of rank $\ell$). Here the center turns out to be just a polynomial algebra in $\ell$ indeterminates, closely related to the enveloping algebra of a Cartan subalgebra and characterized as a suitable algebra of Weyl group invariants. In turn, the "central characters" are easily described and help to sort out representations (as in the earlier use of isolated Casimir elements) even in the more complicated infinite dimensional setting of the BGG category. The notion of "infinitesimal character" even plays a role in the study of Lie group representations, but is only one of many ingredients there. Note that trace functions arising from representations are (as in Drinfeld's construction) a natural way to relate the center to the representation category.
In prime characteristic things get more complicated, as in Lusztig's work on quantum groups at a root of unity: here the finite Weyl group tends to give way to an infinite Coxeter group such as the affine Weyl group (of Langlands dual type) and much is still not understood even though the quantum group representations have been fairly well sorted out. Along the way the role of the center gets diminished, though is still potentially quite interesting.
If the question here is limited just to the Lusztig version of quantized enveloping algebras in characteristic 0 (for an indeterminate $q$), I'm not sure the center has yet been understood well enough from Drinfeld's viewpoint to contribute much to the study of finite dimensional representations. The latter closely resemble the familiar highest weight representations of a semisimple Lie algebra and can be viewed as "quantizations".
By now there is of course a lot of literature to consult, beyond the original papers by Drinfeld and Jimbo. Jantzen gives a good introduction to Lusztig's theory, while Lusztig's many papers (and one book) go farther. But the centers of the various Hopf algebras remain mysterious, to me at least.
ADDED: Probably the most useful book to consult is the 1994 Cambridge treatise A Guide to Quantum Groups by Chari and Pressley (corrected paperback reprint in 1995), which also has an extensive bibliography. They follow some of Drinfeld's 1989/1990 paper for their general discussion in 4.2A of "almost cocommutative" Hopf algebras. Chapter 10 deals with Lusztig's formulation of the quantum enveloping algebra, with $q$ an indeterminate. Here the work of Marc Rosso brings out the classical-looking role of the center in the study of finite dimensional representations. See especially Rosso's 1990 Ann. Scient. Ecole Norm. Sup. paper (and related Bourbaki seminar talk), both available online at www.numdam.org by doing a quick search for "Rosso".