Given an algebraically closed field $k$ and a finitely generated commutative $k$-algebra $A$, all simple modules over $A$ are 1-dimensional

• What is the analogous statement for symmetric monoidal $k$-linear categories?
• What is the analogous statement for braided monoidal $k$-linear categories?

We can assume the category is Abelian and the product functor is right exact in both variables

I expect something of the sort "any simple (in some sense) module category is equivalent to $Vect$", although I have no idea how braided and symmetric are different in this repsect