Let $\mathbb N = \{0,1,2,\ldots\}$ be the monoid of natural numbers with zero. Suppose $S \subseteq \mathbb N \times \mathbb N$ be some subset such that the number of sets of the form $\{ (i,j) \mid (i+n, j+m) \in S \}$ for fixed $(n,m)$ is finite, so all those sets could be referenced by a finite number of tuples $(n_1, m_1), \ldots, (n_k, m_k)$ (lets call them the translations sets with respect to $S$). Now take the smallest submonoid $M = \langle S \rangle$ generated by $S$. If the family of subsets $$ \{ \{ (i, j) \mid (i + n, j + m) \in M\} \mid n,m \ge 0 \} $$ is finite, could it be somehow bounded in the number of translation subsets of this form for $S$?
Remark: The assumption that the translation sets for $M$ are finite is necessary, as in general it could be infinite, take $S = \{(1,1)\}$, then $M = \{ (0,0), (1,1), (2,2), \ldots \}$ and so we have the translation sets $$ \{ (i, j) \mid i = j \},\quad\{ (i,j) \mid i = j + 1 \},\quad\{ (i,j) \mid i = j + 2 \}, \ldots $$ where the first is with $(n,m) = (0,0)$, the second with $(n,m) = (1,0)$ and so on.