Your question is dealt with (in a slightly more general setting) in section 11 of Daniel Schäppi's paper, Tannaka duality for comonoids in cosmoi. Specializing to your setting, he shows that there is a biadjunction (a weak 2-categorical form of adjunction) between the 2-category of $k$-linear categories equipped with a functor to $\operatorname{FinVect}$, where morphisms are triangles commuting up to specified natural isomorphism, and the usual category of coalgebras (thought of as a 2-category with only identity 2-morphisms). I believe this biadjunction should restrict to a biequivalence on the sub-2-category of Tannakian categories (as you have defined them).

This biadjunction is useful because there is a nice tensor product on the 2-category of $k$-linear categories that turns this biadjunction into a monoidal biadjunction, which gives us a way of relating things like bialgebras and tensor categories.

Your question is dealt with (in a slightly more general setting) in section 11 of Daniel Schäppi's paper, Tannaka duality for comonoids in cosmoi. Specializing to your setting, he shows that there is a biadjunction (a weak 2-categorical form of adjunction) between the 2-category of $k$-linear categories equipped with a functor to $\operatorname{FinVect}$, where morphisms are triangles commuting up to specified natural isomorphism, and the usual category of coalgebras (thought of as a 2-category with only identity 2-morphisms). I believe this biadjunction should restrict to a biequivalence on the sub-2-category of Tannakian categories (as you have defined them).

This biadjunction is useful because there is a nice tensor product on the 2-category of $k$-linear categories that turns this biadjunction into a monoidal biadjunction, which gives us a way of relating things like bialgebras and tensor categories.