Isomorphism of ‘ordered’ DAGs / acyclic semiautomata
If you only need to order the outgoing edges the problem is GI complete. Reduce from GI of directed graphs. Given a digraph $D$ make a new one $D’$ as follows: Make a vertex in $D’$ for every vertex of $D$ and every arc of $D$. For every arc $u \rightarrow v$ of $D$, add the arcs: $uv \rightarrow u$ and $uv \rightarrow v$ in $D’$ (in this order). Clearly $D’$ is an (ordered) DAG and two digraphs $D_1$ and $D_2$ are isomorphic iff their corresponding ordered dags $D_1’$ and $D_2’$ are (ordered) isomorphic.
If you need to order both outgoing and incoming edges there is a simple polynomial time algorithm: if $D_1$ and $D_2$ are connected then fixing $f(u) = v$ (for $u \in V(D_1)$ and $v \in V(D_2)$) uniquely determines $f$.