The group $G$ being simply connected is not a notion that one can detect from the usual universal enveloping algebra $U(\mathfrak g)$, so one shouldn't expect there to be a notion for quantum groups $U_q(\mathfrak g)$ corresponding to $G$ being simply connected. You should think of $U_q(\mathfrak g)$ as just giving the local structure around the identity, and so global notions like being simply connected don't make sense (for exactly the same reason that $G$ and $G/N$ have the same Lie algebra for any finite normal subgroup $N\lhd G$).
An "intrinsic" perspective is as follows. Let $\hat{\mathcal O}_{G,e}$ be the complete local ring of $G$ at the identity $e\in G$. Then $U(\mathfrak g)$ is a certain dual of $\hat{\mathcal O}_{G,e}$ (it's the direct limit of $(\hat{\mathcal O}_{G,e}/\mathfrak m_{G,e}^n)^\ast$). Thus $U(\mathfrak g)$ only sees the local group structure in a formal neighborhood of $e\in G$.
There may indeed, however, be a natural notion of being simply connected if you look at the models of quantum groups like $\mathbb Z[q,q^{-1}]\langle a,b,c,d\rangle/(ab-qba,bc-cb,cd-qdc,\text{etc})$ a,b,c,d\rangle$ modulo the relations $ab=q^{-1}ba$, $ac=q^{-1}ca$, $cd=q^{-1}dc$, $bd=q^{-1}db$, $bc=cb$, $ad-da=(q^{-1}-q)bc$, and $ad-q^{-1}bc=1$ (corresponding to the classical group $\operatorname{GL}(2)$).\operatorname{SL}(2)$).
