Why are coproduct objects corepresentations of cartesian products, rather than representations of disjoint unions?

Suppose we had, for some objects $A$ and $B$, an object $C$ such that $(C,-)$ is naturally isomorphic to $(A,-)+(B,-)$. Then the category in which this happens cannot have a terminal object, since such an object, call it $T$, would have only one element in $(C,T)$ but two in $(A,T)+(B,T)$.

More generally, any functor of the form $(C,-)$ preserves all the products that exist in your category. So $(A,-)+(B,-)$ would have to preserve products, and this leads to lots of problems since $(A,-)$ and $(B,-)$ individually also preserve products. More generally yet, you'll have a problem with preservation of limits.

I'm not sure how far one can carry this line of reasoning, but it certainly prevents the existence of such representing objects $C$ in the categories that people usually want to work with. (At the moment, I can't think of any category where such $A,B,C$ exist, but that may be just a deficiency of my imagination.)

Coproducts don't always look much like disjoint unions; consider the coproduct of groups, namely the free product. More formally, there's a condition called disjointness that coproducts can satisfy, and they don't always satisfy it.

You can think of your definition as an attempt to formalize disjointness; unfortunately it doesn't work. You're asking for an object $D$ such that a morphism $f : C \to D$ is either a morphism $C \to A$ or a morphism $C \to B$. Unfortunately, even disjoint unions don't satisfy this property! You don't expect this kind of thing unless $C$ is connected in some sense (say, it is a connected topological space in $\text{Top}$ or a connected graph in $\text{Graph}$); in general some part of $C$ might map to $A$ and some other part can map to $B$.

Nevertheless there is something to this idea; it's one way to try formalizing a sum type, namely a type whose terms are either terms of some type $A$ or some other type $B$. From this perspective it's not obvious that sum types have anything to do with coproducts; I wish I understood this better than I do.

A definition that seems related to bridging this gap is the notion of an extensive category.