A category with a monoidal structure provided in this way is sometimes called a (co)cartesian category, although this is also sometimes used to mean a finitely complete category. (See https://ncatlab.org/nlab/show/cartesian+category.)
As this works with either products or coproducts, we don't set up either construct as an instance.
A category with finite products has a natural monoidal structure.
A category with finite coproducts has a natural monoidal structure.