Let $|R|=|S|=\infty$. In many concrete categories, I know $R^S$ can be identified as the set of all functions from S to R, and the much "smaller" $R^{\oplus S}$ can be identified as the subset of $R^S$ containing only elements with finitely many non-null coordinates. I am looking for a deep reason why this is so across these categories.
In topology, the distinction is obvious, and the product topology is defined as the coarser of the two. However, this is not a deep or fundamental reason as to why $R^{\oplus S}$ has the above characterization in algebra, category theory, or set theory. For example, using just the universal properties of the product and coproduct, it is not at all obvious to me why $R^{\oplus S}$ is characterized the way it is above.
If it is simply defined this way, then why? If it can be derived, then may I see such a derivation?
It is because the operations in the structures are defined to be finitary.
E.g., for Abelian groups, as you said, $R^{\oplus S}$ should satisfy the universal property of coproduct, so this should consist of the $|S|$ copies of $R$, and everything else that is needed to generate the given structure, which are the formal sums of these elements in this case.
But the very definition of Abelian group (module, ring, etc.) only requires a finitary addition operation, so these formal sums, required by the structure, suffice to stay finite.
Note that, in $\mathcal Top$ the $S$-fold coproduct of $R$ is rather the disjoint union of $|S|$ pieces of $R$.