It is true. You can use the path optimality condition for a minimum spanning tree.
Path optimality condition: Let $G$ be a connected undirected network with edge costs $c_{ij}$. Then a spanning tree $T^*$ is a minimum spanning tree if and only if, for every edge $(k,l)$ not in $T^*$, $c_{ij} \leq c_{kl}$ for every edge $(i,j)$ on the path in $T^*$ connecting nodes $k$ and $l$.
Since $M$ is an MST for $G$, we know path optimality holds for every edge in $G \cup e$ not in $M$ except possibly for $e$. Thus determining whether $M$ is an MST in $G \cup e$ is equivalent to checking path optimality for $e$. But since $e$ is the only edge in $M \cup e$ not in $M$, determining whether $M$ is an MST for $M \cup e$ is also equivalent to checking path optimality for $e$. Thus $M$ is an MST in $G \cup e$ iff $M$ is an MST in $M \cup e$.