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.)
The tetrahedron can be vertex inscribed into the cube (just alternating the cubes vertices). Thus the tetrahedrons edges become diagonals of the cubes faces. Therefore each edge of the tetrahedron uniquely corresponds to a face of the cube. This connection can be indeed given by their respective midpoints.
The cube on the other hand can be vertex inscribed into a dodecahedron. This best can be seen when attaching on the cubes faces hipped roofs within alternating orientations, where the trapezia and obtuse triangles pairwise will become reconnected (across the cubes edges) into the required pentagons. Thus again you'll have a unique correspondance from the edges of the cube to the faces of the dodecahedron. But, in contrast to the above case, the edges of the cube do not run through the centers of the pentagons. Therefore the respective centers do not align here.
--- rk