This look like it would bandage pretty quickly. Is that due to the underlying (mystery) geometry or by design?
The geometry jumbles, which explains its apparent bandaging. Compared to other jumbling geometries, its bandaging is very mild. People who can solve the 3x3x3 should not have too much difficulty in solving this puzzle. It is more work and logistics, but not fundamentally harder.
Why should every arbitrary polyhedron have a special name?
To distinguish it from any other arbitrary polyhedron. Either, someone else has discovered this geometry before. In that case, I prefer to reuse the reference to that earlier discovery instead of coining a new, redundant name. For lack of such a reference, this geometry is tentatively called "Jumble Pebble".