Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

That the only monoidal deformations of the category of representations of $U(\mathfrak{g})$ is the category of representations of $U_q(\mathfrak{g})$ is known in Type A from Kazhdan-Wenzl (Adv. Soviet Math, 1993), for the category of vector representations in Type BCD by combining results of Wenzl-Tuba with Liu, and for $G_2$ by Kuperberg. It is open for the other exceptional Lie algebras. I think it's open in type BD if you include the spin representations, though Wenzl has some partial results.

$\newcommand{\g}{\mathfrak g}$ I think the statement Scott Carnahan was refeering to in his answer concerns in fact formal deformations of representations of $\g$, i.e. deformations over the ring $\mathbb C[[\hbar]]$. In that case the reference is Drinfeld's "On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\bar{\mathbb Q}/\mathbb Q)$". If I remember correctly, it says more generally that for any finite-dimensional Lie algebra there is a bijection between equivalence classes of such deformations as braided monoidal categories and $S^2(\g)^\g[[\hbar]]$, and if $\g$ is simple this is one dimensional. I think his results also implies that in the simple case those are the only deformations as monoidal categories as well (this is definitevely true, I'm just unsure if it can be extracted from his paper directly).

There is a nice interpretation of this and of similar results in the framework of shifted Poisson geometry, I highly recommend having a look at Pavel Safronov's very cool paper "Poisson-Lie structures as shifted Poisson structures" (arXiv:1706.02623). In particular this basically settle the question for monoidal deformations and arbitrary finite-dimensional $\g$.

Edit: My first answer was claiming something wrong in the general case, I corrected.

The following is not really an answer but a rather too-long comment, with respect to your second question:

Does there exist deformations of the monoidal category of $U(\frak{g})$-modules which are not equivalent to the cateogry of modules of $U_q(\frak{g})$, for some $q$?

2-parameter deformations of the universal enveloping algebras $U_{pq}(\frak{g})$ have been studied such as for example: $U_{pq}[sl(2)]$, $U_{pq}[sl(3,C)]$, $U_{pq}[u(2)]$, $U_{pq}[u(1,1)]$, etc. Most of these are based on the two-parameter deformation function $$ [x]_{pq}=\frac{q^{x}-p^{-x}}{q-p^{-1}} $$ I am not really sure as to whether the representation categories of such deformed algebras are braided or even monoidal or if they may be described as deformations of the symmetric monoidal categories of the representations of $U(\frak g)$.
However, in the first of the above references, the authors claim that:

the two-parametric quantum group denoted as $U_{pq}[sl(2)]$ admits a class of infinite-dimensional representations which have no classical (non-deformed) and one-parametric deformation analogues, even at generic deformation parameters

which may be considered -in my understanding- as an indication that the representation categories of $U_q[sl(2)]$ and $U_{pq}[sl(2)]$ cannot be equivalent.
Thus, if i am correct in this and given the excerpt from the OP, the categories of $U_{pq}[sl(2)]$-modules cannot be braided.
Also, given Noah Snyder's answer, such categories of representations of two-parameter deformed algebras cannot even be monoidal.

On the other hand, in the last of the above cited articles, a $pq$-deformed coproduct $\Delta_{pq}$ is given for $U_{pq}[u(2)]$ and an element $R_{pq}$ such that: $$ \Delta_{qp}=R_{qp}\Delta_{pq}R_{qp}^{-1} $$ For $p=q^{-1}$, $R_{qp}$ reduces for the universal $R$-matrix of Drinfeld. However, i do not generally know if categorical implications (such as an associated braiding or deformed braiding) have been explicitly studied for $R_{qp}$ and more generally for deformations of this kind.