In a (unit) ring, elements (of a set $S$) are able to operate on each other via $\cdot,+$. If we are to consider the maps $M:S\times S\to S:(a,b)\mapsto a+b$ and $G:S \times S \to S: (a,b )\mapsto a\cdot b$, we have that $\left \langle S,M \right \rangle$ is a monoid and $\left \langle S,G \right \rangle$ is a group. If we define the composition of maps in $G$ and $M$ such that they are distributive, we have a unit ring.
It is clear that many abstract-algebraic objects (fields, commutative rings, etc.) can arise from taking the 'product' of two other objects in a method similar to above.
Is there a category-theoretical way to view the 'product' of two abstract-algebraic objects in a general sense? Are there any applications of this?
To some extent the answer is yes. There is a structure known as an operad, or a multicategory, which is basically just like a category only morphisms may have tuples of objects as domains, instead of just a single object. Any category with a monoidal product gives rise to an operad simply by considering the analogue of functions of several variables.
All those operads form a category, an extension of the category of categories (ignoring size issue, otherwise insert 'small' where needed). The category of operads has a very interesting and quite complicated tensor products known as the Boardman-Vogt tensor product. With respect to that tensor product the category of operads is closed.
Now the fun begins. For many algebraic structures $p$, such as monoids, commutative monoids, magmas (but not for all algebraic structure, e.g., groups) there is an operad $P$ such that the internal hom object $[P,Q]$ models the operad (and thus a category plus more structure) of all $p$-structures in $Q$. In particular, there is an operad $As$ such that $[As,Set]$, the latter viewed as an operad via the cartesian product of sets, is precisely the operad of associative monoids. Similarly, there is an operad $Comm$ for commutative operads etc.
The Boardman-Vogt tensor product gives a way to construct operads modeling $p$-structures in $p'$-structures, simply because $[P,[Q,R]]\cong[P\otimes Q,R]$. For instance, $[As,[As,Set]]$ models associative monoids in associative monoids, that is, a set with two compatible monoid structures. And this is essentially the same as $[As\otimes As, Set]$. It can be shown (this is not hard) that $As\otimes As\cong Comm$, showing that associative monoids in associative monoids are commutative monoids.
The situation gets much more complicated. See for instance Fiedorowicz and Boardman. For a slow-paced intro I shamelessly recommend From Operads to Dendroidal Sets.