The monoid is the set of all sets of integers (but reals or complex numbers could work too). Addition between two elements is defined as $a+b = \{\ x+y\ |\ x \in a,\ y \in b\ \}$. As far as I can tell, the only category of algebraic structures this fits into is monoids, since it's not a group.
Is this structure known or used anywhere, and does it have a name?
This operation is called Minkowski addition. I'm not aware of any special name for the algebraic structure of $\mathcal{P}(\mathbb{Z})$ (or whatever) equipped with Minkowski addition though.