As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation $$ [E,F] = \frac{K-K^{-1}}{q-q^{-1}}. $$ To address this problem, one has an alternative formulation of $U_q(\frak{sl}_2)$: One removes the offensive relation, introduces an extra generator $G$, along with extra relations, $$ [G,E] = E(qK+q^{-1} K^{-1}), ~~~~~~~~~ [G,F] = -(qK + q^{-1} K^{-1})F, $$ and $$ [E,F] = G, ~~~~~~~~ (q-q^{-1})G = K - K^{-1}. $$ and canonical extensions of the Hopf algebra maps. This new algebra can be shown isomorphic to $U_q(\frak{sl}_2)$, yet is well-defined at $q=1$. It is not equal to $U(\frak{sl}_2)$, but to a double cover of $U(\frak{sl}_2)$.

What happens in the general $U_q(\frak{g})$ case? Again $U_q(\frak{g})$ is not well-defined since we the relation $$ [E_i,F_j] = \delta_{ij}\frac{K_i-K^{-1}}{q-q^{-1}}. $$ Judging by the representation theory, there should exist a reformulation of $U_q(\frak{g})$, where the offensive relations are removed, $n$ new generators $G_i$ (where $n$ is the rank of $\frak{g}$) are introduced, along with an extra set of new relations, which I would guess as $$ [G_i,E_j] = \delta_{ij}E_i(qK_i+q^{-1} K_i^{-1}), ~~~ [G_i,F_j] = -\delta_{ij}(qK_i + q^{-1} K_i^{-1})F_i, $$ and $$ [E_i,F_j] = \delta_{ij}G_i, ~~~~~~~ (q-q^{-1})G_i = K_i - K_i^{-1}. $$ This should then be equal to $U_q(\frak{g})$, and is clearly well-defined when $q=1$. I would then guess that at $q=1$ we have an $n$-fold cover of $U(\frak{g})$, which one should consider as justification for the $2^n$ representations types of $U_q(\frak{g})$. Am I OK in reasoning here?