Actually, the category of motives isn't equivalent to the category of representations of an affine group scheme except in characteristic zero, and even there the equivalence depends on the choice of a fibre functor.

The functor to motives is supposed to be a universal cohomology theory. Certainly, one would like the target of a cohomology theory to be at least tannakian.

If you assume the Hodge conjecture, then the affine group scheme attached to the category of abelian motives (that generated by abelian varieties) is more-or-less known --- at least its algebraic quotients are classified.

In the general case nothing is known except that the group is VERY BIG --- for example, over $\mathbb{C}$ it has uncountably many distinct quotients isomorphic to PGL(2).

Actually, the category of motives isn't equivalent to the category of representations of an affine group scheme except in characteristic zero, and even there the equivalence depends on the choice of a fibre functor.

The functor to motives is supposed to be a universal cohomology theory. Certainly, one would like the target of a cohomology theory to be at least tannakian.

If you assume the Hodge conjecture, then the affine group scheme attached to the category of abelian motives over $\mathbb{C}$ (that generated by abelian varieties) is more-or-less known --- at least its algebraic quotients are classified.

If you assume the Tate conjecture, then the affine groupoid attached to the category of motives over an algebraic closure of $\mathbb{F}_p$ is more-or-less known.

In the general case nothing is known except that the group is VERY BIG --- for example, over $\mathbb{C}$ it has uncountably many distinct quotients isomorphic to PGL(2).