People often speak about multiplicative groups or additive groups referring to the operation that is defined on the set of interest. My question is: are there some intrinsic properties of these operations (other than those associated with the group structure) that let one to qualify a operation as a "multiplication" or an "addition"?. In other words: is there any structural definition of "addition" or "multiplication"? . If both meet group properties, what else tells one from the other from a structural point of view?
No, people consider an operation to be an addition or a multiplication according to their own intuition about the structure. Sometimes one is more natural to the other, and people choose accordingly.
There are still some kind of general conventions : an addition is always commutative (it's very bad practice to use $+$ to denote a non-commutative operation), and an operation that is some kind of "function composition" (in a vague and large sense) is generally multiplicative.