For a closed smooth $n$-manifold ($n\ne 4$), the Lipschitz structure is unique by the result of Sullivan. How about the Alexandrov spaces?
At first I was thinking it is trivially true by induction, since locally distance ball in an Alexandrov space is homeomorphic to cone over it's space of directions, which is also an Alexandrove space. But for 5-dimension and higher, the space of directions could be 4-dimensional Alexandrov spaces which admits a metric with curvature bounded from below by 1.
Now one can ask another question for manifold: "whether there exists a positively curved 4-manifold that admits two different Lipschitz structure?"