This is called a multiplex, multi-layer graph or complex network (depends where you look in the literature).

See for instance (1) (in particular the introduction), for more references on complex networks.

(1) Node and layer eigenvector centralities for multiplex networks. F Tudisco, F Arrigo, A Gautier - SIAM Journal on Applied Mathematics, 2018 (arXiv)

On page 4 of 1 , you find:

[...] a multiplex network can be represented as a collection of graphs $$\mathcal{G}=\{G^{(\ell)}=(V_n,E^{(\ell)})\}_{\ell \in V_L}$$ where $V_n=\{1,\ldots,n\}$ is the set of nodes, $V_l=\{1,\ldots,L\}$ s the set of layers and $E^{(\ell)}\subset V_n\times V_n$ is the set of edges on layer $\ell$.

These structures are also sometimes called multi-layer graphs. Note that in the above formulation it is assumed that the nodes on each layer are the same (this models the bijective edges you are referring to).

1 Node and layer eigenvector centralities for multiplex networks. F Tudisco, F Arrigo, A Gautier - SIAM Journal on Applied Mathematics, 2018 (arXiv)