Post correspondence problem for finite monoids
Yes, it is decidable.  Build a graph where each vertex is a pair $(r,s)$ of elements from $M$.  Add all edges of the form $(r,s) \to (r m_i, s m'_i)$ for all $r,s,i$.  Then, your question asks whether there exists a path in this graph from the vertex $(1,1)$ to any vertex of the form $(t,t)$.  This can be answered using standard reachability algorithms (e.g., DFS).  The running time is linear in the size of the graph (i.e., $O(|M|^2 n)$), so the problem is decidable.