What is the significance of multiplication (as distinct from addition) in algebra & ring theory?

In higher math, operators are defined over a set of objects; and these operators are usually denoted as addition and multiplication with a distribution rule. Assuming multiplication is not repeated addition -- though it can be thought that way in some contexts -- what are the needs that push us to define two operators like this (say, if we hadn't learned the usual multiplication already)?

Why not ten operations? Does anything more complicated always reduce to just two operators?

In some areas, multiplication is sometimes a cross product or even a bracket/commutator. Is there some underlying idea here that "multiplication" represents? If so, what is the principle that causes us to generalize?

I too wondered about this when first introduced to rings.  Evidently such structures do exist.  Some examples include bilattices and operads (which I've shamelessly lifted from here).

Another generalization in a different direction is $n$-ary groups. An $n$-ary group has only one operation, but it is defined as a function from $G^n\rightarrow G$.  So, groups in the usual sense would be called $2$-ary groups.

Regarding multiplication: in ring and field theory, we usually call the second operation "multiplication" simply because the most common rings are $\mathbb{Z},\mathbb{Q},\mathbb{R},$ and the like, in which the multiplication operation is actually multiplication in the classic sense.  However many rings involve operations which have nothing to do with the standard idea of multiplication.

The most immediate examples that come to mind are rings of functions, in which addition is be defined pointwise - that is, $(f+g)(x)=f(x)+g(x)$ - and multiplication is defined by composition - that is, $(fg)(x)=f(g(x))$. It is simply an issue of notation that we write $fg$ instead of $f\circ g$ in this context, perhaps to encourage a metaphor with abstract rings, so that we are reminded that all the standard ring theory results still apply to rings of functions.

For an even weirder example of multiplication, if you're savvy on groups, you might like reading about group rings, which consist of "polynomials" in elements of a group $G$ with coefficients in a ring $R$. Multiplication in the group ring is defined by combining the multiplicative group operation of $R$, the group operation of $G$, and the "polynomial" structure of its elements into one big operation.

Widespread utility: many interesting and useful structures have operations satisfying the group or ring axioms, and there are a lot of useful techniques for studying them, especially for special classes of them.

Sometimes, it is even useful to invent a group or ring to describe a situation where no such structure already exists, simply because we know how to study them.

In other words, we study ring theory not because there is some deep idea that all algebraic structures are groups and rings, but because there is a lot that we can say about them, and we can use that knowledge a lot.

Whenever a binary operation is associative, whether it is commutative or not, it can be "represented" as an operator, with the binary operation corresponding to "composition."

For example, let $(G,\times)$ be associative. For each $g\in G$, define $\phi_{g}:G\to G$ as $\phi_g(h)=g\times h$. Associativity can then be written as:$$\phi_g\circ \phi_h = \phi_{g\times h}$$.

Indeed, in general, $(X,\times)$ is associative if and only if the map $X\to X^X$ taking $x\to \phi_x$ is a homomorphism of $(X,\times)$ and $(X^X,\circ)$ as sets with binary operators.

It is because of this relationship between associative operations and function composition that we often write binary operations as multiplication - the fact that we often write $\phi_1\phi_2$ for $\phi_1\circ\phi_2$.

This "representation:" $$(X,\times)\to(X^X,\circ)$$ is not necessarily faithful. It is possible for $\phi_x=\phi_y$ with $x\neq y$. But most of the cases of associative operators that you run into in the real world are faithful under this representation. In particular, whenever the binary operation has an identity element, this representation is faithful.

Indeed, we often get restrictions. Multiplication in a ring, for example, has the distributive law, which essentially means that the $\phi_r$ are homomorphisms of the abelian group $(R,+)$. So here, the representation is $(R,\times)\to (Hom_{\mathcal {Ab}}(R,R),\circ)$. Or for groups, $\phi_{g}$ are always $1-1$ and onto. That "subset" relationship - what collection of "maps" from $X$ to $X$ do we allow for the $\phi_x$ - is often the defining characteristic of our class of algebras.

Multiplication (or, rather, using multiplicative notation) is the "default" binary operation used when speaking of groups and the operations defined on those groups. The actual operation may not be simple multiplication, as we use in arithmetic (the operation may be a permutation, the composition of functions, matrix multiplication, the direct product, etc.).

Using multiplicative notation allows for a more concise description of what, e.g., a group is, regardless of the set in question or the operation on the set, without getting bogged down in the details of trying to define what operation is being used in which contexts, and it provides a consistent notation for generalizing about groups.

For example, exponentiation of a group element simply represents the repeated application of a group's binary operation on a given group element with itself.

Typically, but not always, additive notation is used when generalizing about an abelian group, or to represent a binary operation that is commutative in larger structures.

Perhaps I am not understanding your question. But I think using multiplicative notation (not necessarily the operation of multiplication, as in arithmetic) is pretty much a matter of convention, convenience, and utility (to facilitate abstraction from the concrete to the general).

Similarly, when discussing the multiplicative operator in other algebraic structures, "generally speaking" (and apart from the appropriate operation in a particular structure), it is largely a notational convention, but it also provides a means to abstract from and characterize the essential properties shared by all group structures, or by all rings, or by all fields, etc..