As far as I understand, you start with a "base set" $V$ and consider the monoid obtained by endowing the power set of the power set of $V$ with the binary operation that sends a pair $(A, B)$ of families of subsets of $V$ to $\{X \cup Y \colon X \in A,\, Y \in B\}$. If so, then you may want to read about power monoids, especially if you are interested in questions relating to factorization (as it seems that you are).
Given a (multiplicatively written) monoid $H$, the full power monoid of $H$ is the monoid $\mathcal P(H)$ obtained by endowing the power set of $H$ with the operation of setwise multiplication induced by $H$, so that $AB := \{ab: a \in A, \, b \in B\}$ for all $A, B \in \mathcal P(H)$. The OP is about the special case where $H$ is the monoid obtained by equipping the power set of a certain set $V$ with the binary operation $(X, Y) \mapsto X \cup Y$. So the answer to the question:
do you know of results relating to the factorizability of sets in this way?
is yes in the sense that this is a special case of a broader class of monoids where people (including yours truly) have studied, and are studying, various properties that can be labelled by the "Factorization" tag.
By the way, product-sets in all sorts of monoids (and, in particular, sumsets in additively written groups) are one of the main objects of interest in arithmetic combinatorics. So, stiching the pieces together, I would say that also the answer to the question:
do people use this operator (if so, what is it called? I don't want to reinvent the wheel)?
is yes (and you are reinventing the wheel).