In Rel, both products and coproducts amount to taking a disjoint union of sets. Is there a category whose object class can be interpreted as the class of all sets, in which both products and coproducts amount to taking Cartesian products, in the classical sense?
In the category of Abelian groups:
If $A$ and $B$ are abelian groups, the product $A\times B$ together with $\iota_A:a \longmapsto (a,0)$ and $\iota_B:b \longmapsto (0,b)$ is a co-product. Since if $f:A \longrightarrow C$ and $B \longrightarrow C$ are morphisms of abelian groups, then $h:A\times B \longrightarrow C$, defined by $(a,b)\longmapsto f(a)+g(b)$, is the unique homomorphism making the required (coproduct) diagram commute.
This is only true for finite-products, but hopefully it answers the question.