In other words, you are interested in the spaces where *isometric subsets are congruent*.

If the metric space is locally compact and intrinsic and simply connected then you get only spheres, Euclidean spaces and hyperbolic spaces.
If not simply-conected, then in addition you get real projective spaces.

If the metric is not intrinsic you get discrete spaces and yet Cantor-like spaces build on them (who knows what else).

Yet in [Urysohn universal space][1] the property holds for compact subsets.

  [1]: http://en.wikipedia.org/wiki/Urysohn_universal_space