Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$, including those that have $0$ columns or $0$ rows. Let the monoid operation be $\oplus$, denoting direct sum. Let $M \sim K$ be an equivalence relation expressing that there exist unitary matrices $U$ and $V$ such that $UMV = K$. The equivalence relation $\sim$ is therefore *unitary equivalence*. Has the monoid $M(R,*)/\sim$ ever been studied in the literature?

Note: For $$(R,*) \in \{(0, \operatorname{id}_{0}), (\mathbb R, \operatorname{id}_\mathbb R), (\mathbb C, a + bi \mapsto a - bi), (\mathbb R[\varepsilon]/(\varepsilon^2),a+b\varepsilon \mapsto a+b\varepsilon), (\mathbb R[\varepsilon]/(\varepsilon^2),a+b\varepsilon \mapsto a-b\varepsilon)\}$$ we have that $M(R,*)/\sim$ is isomorphic to a *free commutative monoid*. In fact, I don't know of any $M(R, *)/\sim$ which is not isomorphic to a free commutative monoid. When such an isomorphism exists, it effectively generalises the Singular Value Decomposition.