quantum mechanics - Why do non-hydrogen atomic orbitals have the same degeneracy structure as hydrogen orbitals?

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

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They have the following degeneracy structure:

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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:

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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?