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I think one key aspect that is overlooked in these replies is the Born-Oppenheimer approximation. The nucleus of any atom can be approximated as a point with very few relativistic correction due to its mass. An electron on the other hand is essentially massless (comparatively speaking. I know it is still a fermion and has mass). Once you put more than one electron in any orbital those relativistic corrections have consequences. So ''hydrogen-like'' means any atom that does not need Pauli or Dirac's insight to explain the deviations in the spectra i.e. the bohr model.