Is there a general term for $A\oplus B = \{a \oplus b | a\in A, b\in B\} $? Is there a general term that specifies that if an operator $\oplus$ is applied to two sets, it's actually applied to all possible pairs of elements of the two sets?
Or is that always the case and goes without saying?
Matrix addition for example is also an elemtent-wise operation, but it does not apply the operator to every possible combination of elements, but only some. How would that way-of-applying-an-operator-to-things-that-have-more-things-in-them be called?
 A: For subsets $A,B$ of some additively written ambient group $(G,+)$ (or also just semigroup) the set $A + B = \{a + b \colon a \in A, \, b \in B \}$ is often called the Minkowski sum. This name is especially common in a geometric context. 
But the concept is very important in Additive Combinatorics and Additive Number Theory. There it is so ubiquitous that it is often simply called sum of the sets $A$ and $B$, or sum-set of $A$ and $B$. And in this context it basically goes without saying that this is meant. Though there are still subtle points for example for summing $A$ with itself one sometimes wants to exclude $a+a$, and denotes this then as $A \hat{+} A $ or $2\widehat{\phantom{o}}A$ so that some brief indication what exactly is meant can still be desirable.
Given your example of matrices, let me mention that given functions $f$ and $g$ the operation of constructing the function $x \mapsto f(x) +g(x)$ is often called  the pointwise sum of $f$ and $g$. And the same for  product.  Recall that a sequence is also a function, and also a matrix or a vector can be thought of as a function (defined on the set of indexes), so this applies to them. In that context then termwise or elementwise can also be used rather than pointwise.
A: Note that this cannot be used for all operators. Consider $A \cup B$, which is certainly not all possible unions of an element in $A$ with an element in $B$, though in ZFC we certainly have the set $\{ a \cup b : a \in A \land b \in B \}$.
