A criterion for the planar graph to have unique dual
First let me mention that the definition of being uniquely embeddable requires ANY graph isomorphism (e.g. just renaming symmetric vertices or any other automorphism permutation) to be (not necessary uniquely) extendable to topological (or combinatorial) one (see Diestel Graph Theory chapter about planar graphs for these definitions) in contrast to usual understanding of this definition that for any two planar embeddings there EXISTS some topological (or combinatorial) isomorphism. Difference between the topological and combinatorial isomorphism definitions is negligible for 2-connected planar graphs (see Diestel book). This particularly means that if we reflect or twist some symmetric graph (as done in the proof of necessity of the aforementioned criterion "subdivision of 3-connected planar graph if and only if condition to be embeddable" in the book "Planar Graphs: Theory and Algorithms") we get topologically non-isomorphic graph in general. Though $K_{2,3}$ graph witnesses that the necessity proof here is wrong but the sufficiency is hold.

What I mean by complex-based isomorphism is just combinatorial one from Diestel book.

If we allow my unusual definition of uniqueness that any two embeddings are topologically isomorphic then there is an example of 2-connected non-subdivisions that are uniquely embeddable.

As for aforementioned paper "Uniquely embeddable planar graphs" the special definition which is given there is tougher (not robust to transposition of symmetric vertices) (but close to as Yuval assumed) than topological isomorphism according to its Theorem 3.5.