It is a basic fact in representation theory of finite groups over complex numbers that the character tables of $Q_8$ and $D_8$ are identical. I believe, this implies that the corresponding categories of representations are equivalent (as tensor categories).

On the other hand, Tannakian Formalism tells us that we can reconstruct a finite group $G$ from its category of representations $\mathbf{Rep}_G$ together with the natural (foregetfull) fibre functor $F_G: \mathbf{Rep}_G\rightarrow \mathbf{Vect}$. Namely, $G$ is canonically isomorphic to the tensor automorphisms of the tensor functor $F_G$.

This implies that the fibre functors $F_{D_8}$ and $F_{Q_8}$ are different. How can one see this explicitly in terms of representations of $D_8$ and $Q_8$?