I'll work over $\mathbb{C}$ below for simplicity, although it can be replaced by an algebraically closed field of characteristic $0$.
Assuming that "fusion subcategory generated by" means what I think it means (tensor products, direct sums, direct summands), here is a general observation.
Lemma: The fusion subcategory of $\text{Rep}(G)$ generated by some reps $V_i$ is $\text{Rep}(G/N)$ where $N$ is the intersection of the kernels of $G$ acting on each $V_i$.
This follows in turn from the standard fact that $\text{Rep}(H)$ is generated as a fusion category (in the above sense) by any faithful representation of $H$; see, for example, this MO question. From here it suffices to show that the intersection of the kernels of $G$ acting on $V \otimes V^{\ast}$ for each irrep $V$ is $Z(G)$.
Recall that the regular representation $\mathbb{C}[G]$ decomposes, as a representation of $G \times G$ (acting on the left and right), as $\bigoplus_V V \boxtimes V^{\ast}$ where $\boxtimes$ denotes the external tensor product. If we restrict to the diagonal copy of $G$ in $G \times G$, we get that the permutation representation coming from $G$ acting on itself by conjugation decomposes as $\bigoplus_V V \otimes V^{\ast}$ where $\otimes$ is now the ordinary tensor product of representations. The kernel of this permutation representation is clearly $Z(G)$, as desired.
