Notice that a Tetrahedron has 6 edges, and a Cube has 6 faces. So lets draw points at the center of those 6 edges, and at the center of the 6 faces.
If we project these points onto a unit sphere, do they turn out to be the same?
This is the chart I'm looking at. It's not easy to eyeball, but I suspect the answer is no.
I also have the same question for other pairs. E.g., the Cube has 12 edges and the Dodecahedron has 12 faces. Do their centers coincide on a unit sphere? And what about the Octahedron edges vs the Dodecahedron faces?
I think those are the only possibilities. Can't really talk about the edges of the Dodecahedron or Icosahedron because there are 30. No Platonic Solid has 30 faces.
(Note: I didn't bother with vertexes because the dual of one Platonic Solid will swap the vertexes and faces, even with the Tetrahedron despite being a self-dual.)