Besides that the theories (étale fundamental group and Tannakian formalism) just formally look alike, there exist actual comparison results between certain étale and Tannakian fundamental groups.

Namely: there is Nori's fundamental group scheme $\pi_1^N(S,s)$, where $S$ is a proper and integral scheme over a field $k$ having a $k$--rational point. It is defined to be the fundamental group of some Tannakian category associated with $S$ (to be precise: The full $\otimes$-subcategory of the category of locally free sheaves on $S$ spanned by the essentially finite sheaves). In the case $k$ is algebraically closed, there is a canonical comparison morphism from Nori's fundamental group to Grothendieck's étale fundamental group, and if moreover $k$ is of characteristic zero this morphism is an isomorphism.

Besides that the theories (étale fundamental group and Tannakian formalism) just formally look alike, there exist actual comparison results between certain étale and Tannakian fundamental groups.

Namely: there is Nori's fundamental group scheme $\pi_1^N(S,s)$, where $S$ is a proper and integral scheme over a field $k$ having a $k$--rational point. It is defined to be the fundamental group of some Tannakian category associated with $S$ (to be precise: The full $\otimes$-subcategory of the category of locally free sheaves on $S$ spanned by the essentially finite sheaves). In the case $k$ is algebraically closed, there is a canonical comparison morphism from Nori's fundamental group to Grothendieck's étale fundamental group, and if moreover $k$ is of characteristic zero this morphism is an isomorphism.

One more comment: The classical Tannaka-Krein duality theorem for compact topological groups (see e.g. Hewitt&Ross vol. II) should presumably be another realisation of the common generalisation you seek.