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The solutions of the Schrödinger equation for hydrogen are the "electronic orbitals", shown in this picture:

enter image description here(source)

They have the following degeneracy structure:

enter image description here(source)

It is often said that atoms in other elements simply have more electrons "filling up" the orbitals in increasing order of energy, as required by the Pauli exclusion principle. This is somewhat confirmed via xray spectroscopy:

enter image description here(source)

But why would non-hydrogen atoms have the same orbital degeneracy structure as hydrogen, if the actual Hamiltonian is different from one atom to another?

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I suspect that the shell theorem plays a part in why this is a working approximation. – dmckee Feb 18 at 17:15

Polyelectronic atoms don't have atomic orbitals - though they are a very useful approximation for describing the properties of polyelectronic atoms.

The 1s, 2s, etc orbitals are solutions for a central potential, and for any smooth monotonic central potential we'll get solutions of this form. The radial part of the orbitals will be different for different central potentials but the angular part is dictated by the spherical symmetry and is the same for all (smooth monotonic) central potentials.

But for any atom with more than two electrons the potential is not centrally symmetric because it includes terms like $1/r_{ij}$ for the interaction between the $i$th and $j$th electrons. This means the hydrogenic orbitals are not solutions.

However because the electrons are delocalised over the whole atom the potential is approximately central. By this I mean that if we take a time average potential the $1/r_{ij}$ terms tend to average out to a central force. In that case we do get hydrogenic type orbital as solutions, but we have to bear in mind that they are approximate solutions. They should be regarded as a useful way of building up the complete electronic structure but they are not themselves real. For example in a lithium atom there is not actually two electrons in a 1s orbital and one in a 2s orbital. There is a single three electron wavefunction that can be approximately decomposed into hydrogenic 1s and 2s orbitals for convenience.

Having said this, for most purposes the hydrogenic orbitals are very good approximations. For example when we are considering atomic spectra we usually describe them as transitions between the hydrogenic orbitals and this works pretty well.

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why with two electrons the potential is still centrally symmetric? – Sparkler Feb 18 at 17:07
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@Sparkler: suppose you're sitting on electron $i$ looking at electron $j$. From your perspective electron $j$ will spend as much time to the left of the nucleus as to the right of the nucleus, and as much time above the nucleus as below the nucleus. So even though the force due to electron $j$ is in lots of different directions if you take a time average the force vector will point towards the nucleus i.e. on average it is a central force. – John Rennie Feb 18 at 17:12
    
so Li is not actually $\mathrm{1s^2 2s^1}$ but rather $\mathrm{1\square^3}$..? – Sparkler Feb 18 at 17:12
    
@Sparkler: yes, though obviously your notation is not standard. We would normally describe the polyelectron state using term symbols. – John Rennie Feb 18 at 17:13
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@Sparkler: yes! Basically, as you increase the atomic number the 1s orbital shrinks inwards. – John Rennie Feb 18 at 17:28

Although John Rennie is right to remind us that when talking about orbitals for a multi-electron atom, it is only an approximation, I would like to point out that the source of the exact degeneracy of the (many-electron) atomic states, it is the isotropy of the atom in the absence of external polarizing fields. This makes the (total) angular momentum J and its projection m, exact quantum numbers, applicable to classification of atomic terms.

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