Suppose we have a commutative monoid ${\mathcal M}=\langle M,\otimes\rangle$ such that the usual divisibility relation $\leq_\otimes$ given by $a\leq_\otimes b\Leftrightarrow \exists c(a\otimes c=b)$ is a partial order, and define LCM and GCD-like operations $\mathop{LCM}_{\mathcal M}(a,b) := \min(l: a\leq_\otimes l\wedge b\leq_\otimes l)$ and similarly for the GCD. Then for the monoids $\langle\mathbb{N}_{\geq0},+\rangle$ (where $\leq$ is the usual partial ordering on $\mathbb{N}$, $\mathop{LCM}_\mathcal{M}(a,b)=\max(a,b)$, and $\mathop{GCD}_{\mathcal M}(a,b)=\min(a,b)$) and $\langle\mathbb{N}_{>0}, \times\rangle$ (where $\leq$ is just integer divisibility and $\mathop{LCM}_{\mathcal M}$ and $\mathop{GCD}_{\mathcal M}$ are just the classic definitions of these functions) we have the identity $\mathop{LCM}_{\mathcal M}(a,b)\otimes\mathop{GCD}_{\mathcal M}(a,b)=a\otimes b$. In fact, if this holds for monoids $\mathcal{M}$ and $\mathcal{N}$ then it holds for their direct product, so in particular it holds for $\langle {\mathbb N}_{\geq 0}{}^n, +\rangle$ and holds for the inverse limit $\langle \mathbb{N}_{\geq 0}{}^{\leq \omega}, +\rangle$ (addition in the latter two cases being componentwise); the latter example is just $\langle \mathbb{N}_{>0},\times\rangle$ seen through the lens of prime factorizations.

Is there a term for commutative monoids that satisfy this condition? Are there any other notable examples that aren't just straightforward generalizations of the above? One simple non-example is $\mathcal{M}=\langle\{1\}\cup\{2n\}_{n\in\mathbb{N}},\times\rangle$; we have $\mathop{GCD}_{\mathcal M}(6,18)=1$ and $\mathop{LCM}_{\mathcal M}(6,18)=36$ but $1\times 36\neq 6\times 18$.

ETA: If I read my definitions correctly then in fact every affine monoid satisfies the condition; is it exactly the affine monoids that do?

(Previously asked at https://math.stackexchange.com/questions/4711532/is-there-a-name-for-this-condition-on-a-monoid with no answer)