No, it is based on applying a lattice onto the faces of a dodecahedron (technically a pentakis dodecahedron). Take a look at https://a5geo.org/examples/teohedron-dodecahedron and other examples on the website.
H3 is based on a dodecahedron it is it the reason the cell areas range so much, the same is true of S2 - but this is based on a cube.
The shapes look a bit wonky when projected onto a map, though, and it may not be as intuitive to reason about as the hexagons that would (mostly) result from subdividing an icosahedron. With a subdivided icosahedron you end up with a regular lattice of shapes that is easier to reason about. I think an icosahedron might be a better fit for an indexing scheme for that reason, despite it's higher mathematical error in approximating the sphere at a given resolution.
I explored a similar idea four or five years ago, without being aware of H3. My goal was to find a compact multi-resolution geospatial height map format. My idea was closer to H3 than to yours, it seems.
H3 is based on a dodecahedron it is it the reason the cell areas range so much, the same is true of S2 - but this is based on a cube.