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If $G$ is an affine algebraic group (for example a finite group), then the category of k-linear cocontinuous symmetric monoidal functors from $\mathsf{Rep}(G)$ to $\mathsf{Vect}_k$ is equivalent to the category of $G$-torsors over $k$. In particular, not every such functor needs to be isomorphic to the identity. For example, if $k'$ is finite Galois extension of k with Galois group $G$, then the functor $F(V) = (V \otimes_{k} k')^{G}$ will satisfy all the axioms you will think to write down, but is not isomorphic to the identity functor.

If $G$ is an affine algebraic group (for example a finite group), then the category of $k$-linear cocontinuous symmetric monoidal functors from $\mathsf{Rep}(G)$ to $\mathsf{Vect}_k$ is equivalent to the category of $G$-torsors over $k$. In particular, not every such functor needs to be isomorphic to the identity. For example, if $k'$ is finite Galois extension of k with Galois group $G$, then the functor $F(V) = (V \otimes_{k} k')^{G}$ will satisfy all the axioms you will think to write down, but is not isomorphic to the identity functor.

If G is an affine algebraic group (for example a finite group), then the category of k-linear cocontinuous symmetric monoidal functors from Rep(G) to Vect_k is equivalent to the category of G-torsors over k. In particular, not every such functor needs to be isomorphic to the identity. For example, if k' is finite Galois extension of k with Galois group G, then the functor $F(V) = (V \otimes_{k} k')^{G}$ will satisfy all the axioms you will think to write down, but is not isomorphic to the identity functor.

If $G$ is an affine algebraic group (for example a finite group), then the category of k-linear cocontinuous symmetric monoidal functors from $\mathsf{Rep}(G)$ to $\mathsf{Vect}_k$ is equivalent to the category of $G$-torsors over $k$. In particular, not every such functor needs to be isomorphic to the identity. For example, if $k'$ is finite Galois extension of k with Galois group $G$, then the functor $F(V) = (V \otimes_{k} k')^{G}$ will satisfy all the axioms you will think to write down, but is not isomorphic to the identity functor.

Oops... misunderstood the question.

If G is an affine algebraic group (for example a finite group), then the category of k-linear cocontinuous symmetric monoidal functors from Rep(G) to Vect_k is equivalent to the category of G-torsors over k. In particular, not every such functor needs to be isomorphic to the identity. For example, if k' is finite Galois extension of k with Galois group G, then the functor $F(V) = (V \otimes_{k} k')^{G}$ will satisfy all the axioms you will think to write down, but is not isomorphic to the identity functor.