A question about the Tannaka-Krein reconstruction of finite groups

I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a finite tensor category together with a monoidal functor to vector spaces. See Theorem 5.3.12 of the Tensor Categories. This reconstruction is compatible with the one for groups, in the sense that the Hopf algebra you recover is $k[G]$. However, for the Hopf algebra one you look at all endomorphisms instead of tensor isomorphisms. If you start with H-mod for a Hopf algebra H and look at $\mathrm{Aut}^\otimes(F)$ you'll end up with the group G of grouplike elements of H, and Rep(G) will not be the same as H-mod because H is not spanned by its grouplike elements.

Although you already have one sufficient answer, I thought I would add here that Etingof and Gelaki, "Isocategorical groups" considered exactly this question. For finite groups, they gave a complete characterization of when two non-isomorphic finite groups can have the same monoidal categories $rep(G)$ (ignoring the symmetric structure). One consequence of their characterization is that all finite groups of odd order, or twice odd order, and all simple finite groups are determined even by $rep(G)$ considered only as a monoidal category.