Condition that the faces of a graph be unique? There is a theorem that if a planar graph is 3-vertex-connected, then it has a unique embedding up to a reflection. (See e.g. here.) This means that its faces and its dual graph are uniquely defined.
Is there a generalization of this result to graphs of higher genus (e.g. a graph that is not planar but can be embedded on the torus)?  Are there simple conditions that guarantee that the minimum-genus embedding is unique?
 A: If you're looking for generalizations of properties of planar embeddings then you definitively should have a look at results involving the face-width of a graph.
Directly answering your question, I believe no equivalent result exists. At least not for general surfaces (I vaguely remember some theorems regarding unique embeddings for the Torus and Projective Plane).
However the following result, due to Thomassen[1], may be close to what you want.
The edge-width of an embedding $\Pi$ of $G$ is the length of the shortest $\Pi$-noncontractible cycle. Large-edge-width embeddings are embedding whose edge-width are larger than the maximum length of a facial walk.
Theorem[1]: Let $G$ be a 2-connected graph that has a LEW-embedding in a surface $S$. Then a cycle $C$ in $G$ is contractible in every embedding of $G$ in $G$ if and only if $C$ is contractible in some embedding of $G$ in $S$. If, in addition, $G$ is a subdivision of a 3-connected graph, then $G$ is uniquely embeddable in $S$.
You can find this result and many other on the 5th chapter of the Graphs on Surfaces book.
[1] Thomassen, Carsten. "Embeddings of graphs with no short noncontractible cycles." Journal of Combinatorial Theory, Series B 48.2 (1990): 155-177.
