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..