Book Embedding Duality of Graphs
To keep things a little cleaner let's assume that the graph has a Hamiltonian cycle that's embedded along the spine so that each cell of the embedding lives within a single page. Also those spine edges are places where multiple cells meet so let's make them vertices as well.

Then, within each page of the book, the dual graph is a forest, with all vertices that are not adjacent to a leaf having degree three or more, and having the spine edge vertices as its leaves. So a book embedding with $p$ pages has a dual graph consisting of $p$ plane forests, sharing a common set of leaves in the same cyclic ordering as each other.

Whether this structure is useful for anything is a different question, of course.

As for your question about generalizations of planar graph decompositions into primal and dual spanning trees: I doubt something like that happens in book embeddings, but it can be generalized to embeddings of graphs onto non-planar manifolds. See my paper "Dynamic generators of topologically embedded graphs" (SODA'03).