It turns out that these polyhedra that have congruent vertices and faces have a name. They are the Noble Polyhedra. If one insists that they also be convex the Noble polyhedra are the regular polyhedra plus the disphenoids mentioned in Douglas Zare's answer.
When one allows intersecting faces, however, new collections turn up, such as the stephanoids, originally studied by Max Brüker:
Max Brückner Uber die gleichecking-gleichflachigen, diskontinuierlichen und nichtkonvexen Polyheder
Nova Acta Leop. 86(1906), No. 1, pp. 1 – 348 + 29 plates.
Images of the plates with pictures of the models.
These shapes are also discussed and further developed by Branko Grünbaum:
Polyhedra with hollow faces
Proc. NATO-ASI Conf. on polytopes: abstract, convex and computational, Toronto 1983, Ed. Bisztriczky, T. Et Al., Kluwer Academic (1994), p 43-70.
Grünbaum's constructions do use generalisations of the definition of polyhedra. For a thorough discussion of these (including having polygons return to the same vertex, and coplanar faces) see the following paper, which also has a discussion of Noble Polyhedra.
Grünbaum, B. Are your polyhedra the same as my polyhedra?
Discrete and Computational Geometry: The Goodman-Pollack Festschrift B. Aronov, S. Basu, J. Pach, and Sharir, M., eds. Springer, New York 2003, pp. 461 – 488.
Interestingly the classification of all Noble Polyhedra is still an open problem.