Equality of polyhedra

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Is a minimal representation for a polyhedron unique? And if so can we use this to prove that two polyhedra are equal (or maybe the same is a better definition).

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It depends what you call "minimal representation".

It exists a canonical representation of any polyhedron, called polygonal canonical schemata (see [1]) which is unique. This representation can be used to test if two polyhedra are "equal", more precisely homeomorphic, i.e. they have the same topology.

[1] Gert Vegter , Chee K. Yap, Computational complexity of combinatorial surfaces, Proceedings of the sixth annual symposium on Computational geometry, p.102-111, June 07-09, 1990, Berkley, California, USA