I haven't thought about this stuff in a while, but here's what I can remember. (Perhaps I should just read my notes for a talk on the subject instead.) Before I begin, let me emphasize that nothing I'm about to say is due to me --- much of it dates from my mathematical infancy.
Let me answer your questions a bit out of order.
Another line of thought is that Chern-Simons theory is canonically associated to $\mathfrak g$. Does this lead to a canonical construction of the representation theory of $U_q(\mathfrak g)$? This would appear to work only for q a root of unity.
My memory is that the category $\mathrm{Rep}^{\mathrm{fd}}(U_q\mathfrak g)$ of finite-dimensional $U_q\mathfrak g$-modules can be canonically associated to any semisimple Lie algebra $\mathfrak g$ without extra data, in a purely algebraic way. I don't remember how to do this, at least not slickly. When $q = e^\hbar$ is in an infinitesimal neighborhood of $1$, there is a method, probably due to Drinfeld, which works both for $\mathrm{Rep}^{\mathrm{fd}}(U_q\mathfrak g)$ and for the category of all representations $\mathrm{Rep}(U_q\mathfrak g)$. More generally, it works for any Lie algebra over a characteristic-$0$ field $\mathbb K$ with a chosen nondegenerate adjoint-invariant symmetric pairing (indeed, only a quadratic Casimir is needed, possibly degenerate). For semisimples, you can take the Killing form.
Here's the coarse outline. Take the category $\mathrm{Rep}_\hbar(U\mathfrak g) = \mathrm{Rep}(U\mathfrak g) \otimes_{\mathbb K} \mathbb K[[\hbar ]]$, which is the Kelly–Deligne tensor product of $\mathrm{Rep}(U\mathfrak g)$ and $\mathrm{Rep}(\mathbb K[[\hbar ]])$, along with its standard monoidal functor $\otimes : \mathrm{Rep}_\hbar(U\mathfrak g) \otimes \mathrm{Rep}_\hbar(U\mathfrak g) \to \mathrm{Rep}_\hbar(U\mathfrak g)$, but not its standard monoidal structure — I want to modify the associator, and also the braiding. The braiding is modified in some standard way, and the associator is modified by some $O(\hbar)$ term depending on a choice of Drinfeld associator. These is the main topic of the notes I linked above, so I won't go into it now.
Anyway, this category determines the Hopf algebra $U_q\mathfrak g$ (again with $q = e^\hbar$) up to isomorphism, but not up to canonical isomorphism. That was what I was trying to explain in the answer you linked to.
You are right that Chern–Simons theory also should give a canonical description of $\mathrm{Rep}(U_q\mathfrak g)$, at least when $q$ is a root of unity. Actually, you should choose a compact group $G$ integrating $\mathfrak g$. Then it's expected, but in general not yet proved, that the semisimplified version of $\mathrm{Rep}(U_q\mathfrak g)$ is described by the fusion rules for positive-energy representations of the loop group of $G$ (at a level determined by the order of $q$). This latter category is the "path integral / analytic / Chern–Simons" quantization of $\mathfrak g$.
The idea (as I understand it) is that we should put $q=\exp(h)$ and expand in power series over $h$. Then taking the degree zero part in h gives the representation of 𝔤 and the linear part in h gives some additional structure. If we think of a K-linear category as a K-algebra then this additional structure would be a Poisson structure. This would also be a replacement for Drinfeld's Poisson-Lie groups and Lie bialgebras. Then there would be further structure corresponding to the quasi-triangular structure.
You can think of the "infinitesimal braiding" determined by the Casimir as being analogous to a Poisson structure. Alternately, you can think of the Casimir itself as analogous to a Poisson structure. This has been made precise in the notion of "shifted symplectic structure" prevalent in the AKSZ literature (see e.g. work by Roytenberg). The category $\mathrm{Rep}(U\mathfrak g)$ "is" the category of quasicoherent sheaves on a "stack" $B \mathfrak g$, the "classifying space" of the formal group determined by $\mathfrak g$. I put a bunch of words in quotes because you should think of formal groups and their classifying spaces as infinitesimal objects (in particular, $B \mathfrak g$ is not a stack in the usual algebrogeometric meaning). The Killing form defines a "shifted symplectic structure" on $B\mathfrak g$, meaning a nondegenerate closed 2-form of some cohomological degree (remember that the tangent bundle to a stack is a bundle of chain complexes, not of vector spaces).
That said, there are some open questions relating general shifted-Poisson structures to structures on the category of quasicoherent sheaves.
The answer also made a comment to the effect that the category of (finite dimensional) representations of $U_q(\mathfrak g)$ is a deformation quantisation of the category of (finite dimensional) representations of $\mathfrak g$. My question is whether this comment can be made precise?
It is probably worth emphasizing, earlier than I am doing it, that for generic $q$ (or $q = e^\hbar$), $\mathrm{Rep}(U_q\mathfrak g)$ and $\mathrm{Rep}(U\mathfrak g)$ are equivalent as categories — the only thing deforming is their braided monoidal structures. Actually, I should be a bit careful: there can be multiply non-isomorphic equivalences, and depending on the context either one has been chosen or it hasn't. Compare the case of deformations of Poisson to associative algebras: we usually assume that the Poisson algebra $A_0$ and the deformation quantization $A_\hbar$ are equal as vector spaces (perhaps up to tensoring with $\mathbb K[[\hbar]]$), and that that equality is part of the data. Changing that equality by something $O(\hbar)$ is a "gauge transformation" in Kontsevich's sense.
On the other hand, extracting the Poisson structure doesn't require knowing the equality $A_\hbar = A_0 \otimes_{\mathbb K} \mathbb K[[\hbar]]$, just that $A_\hbar$ is a flat deformation (i.e. $A_\hbar$ is a sheaf of algebras over the formal disk $\mathrm{Spec}(\mathbb K[[\hbar]])$, and I want to know that as a sheaf of vector spaces it is a vector bundle). Similarly, you can extract the infinitesimal braiding just from $\mathrm{Rep}(U_q\mathfrak g)$. I state this more precisely in the "Proposition" in section 2 of my linked lecture notes.
But, in any case, yes: the point is that you can present (in many ways) the category $\mathrm{Rep}(U_q\mathfrak g)$ as having as its objects and morphisms the objects and morphisms of $\mathrm{Rep}(U\mathfrak g)$ (up to base changing), but with its algebraic structure (composition, associator, braiding, etc.) deformed.
