Assume that a digraph is finite and strongly connected, and that all edges and vertices bear labels from some set. Let $f(v)$ be the label of vertex $v$, and $f(e)$ be the label of edge $e$. Say that a path $v=(v_1, e_1, v_2, e_2, ... , e_k, v_{k+1})$ is not unique if there exists a distinct path $v'=(v_1', e_1', v_2', e_2', ... , e_k', v_{k+1}')$ (distinct meaning $v'≠v$) with matching labels, i.e., such that $f(v_i)=f(v_i')$ for $i=1...k+1$ and $f(e_i)=f(e_i')$ for $i=1...k$. A path is unique if there is no distinct path with matching labels.
It's obvious that a graph with a unique path has no non-trivial automorphism (i.e., a non-trivial mapping that preserves labels). Is the converse true? That is, does a finite labeled digraph with no non-trivial automorphism always have a unique path? If the answer is not known, are there any interesting results relevant to this question?