I think it is more or less an accident when two digraphs have the same number of simple cycles, because the number of non-isomorphic digraphs is vastly greater than the number of cycle counts. So unless your two digraphs have some other relationship I doubt that there is any general way to make a useful bijection.

I think it is more or less an accident when two digraphs have the same number of non-simple cycles, because the number of non-isomorphic digraphs is vastly greater than the number of non-simple cycle counts. So unless your two digraphs have some other relationship I doubt that there is any general way to make a useful bijection.

Namely, since the eigenvalues are less than $|V|$ in magnitude, the number of non-simple cycles (more commonly called closed walks) of length at most $|V|$ is bounded by $|V|^{|V|+2}$. However the number of digraphs is around $2^{|V|(|V|-1)}/|V|!$ even if loops are forbidden, which is vastly larger.