The spherical harmonics represent the analytic solution to the Schrodinger equation for a hydrogen in a vacuum. Any other system (e.g. Hydrogen with two electrons, or a neutral Helium atom) does not have any analytical solution (its a many body problem).

In general, determining the wave function for a collection of electrons is a very hard problem. Depending on the accuracy needed, different commonly used methods range from O(n^4) to O(n^7) where n is the number of electrons.

The article is referring to confined free electrons, no nucleus and therefore no orbitals involved.

In other words: different charge distribution, different potential, diffetent resulting wave functions.

Platonic? It isn't exactly the spherical harmonics, because of the radial term, relativistic corrections and stray electromagnetic fields that are always around.

That's what makes it Platonic -- there is no perfect sphere, triangle, etc... But material substances approximate spheres, etc as a platonic form.

Also "platonic", because their school of thought held that the properties of matter were based on the geometries of their component atoms, which they assumed to be the simplest forms possible (platonic regular forms). They were wrong on the specifics (atoms aren't made of tetrahedra or cubes), but the basic atomic theory (atomic geometry determines functional properties) remains an essential insight.