What is a "hydrogen-like" or "hydrogenic" atom?
A hydrogen-like atom (or ion) is simply any particle with a nucleus and one electron.



That should be sufficient to answer the question at hand, but I thought I should say a bit more, as some of these answers are potentially confusing.

The historical reason why the Rydberg formula only works for hydrogen-like atoms is because it was originally formulated to explain the spectral lines of hydrogen. It was never intended to explain the spectra of multi-electron atoms.

The physical reason, though, is because the Rydberg formula uses energy levels that only depend on the principal quantum number $n$, which has to be a positive integer:

$$\bar{\nu} = Z^2\mathcal{R}\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \qquad n_1,n_2 \in \mathbb{Z}^+$$

and nowadays we know that this is only true for hydrogen-like atoms;$^*$ the energy levels of multi-electron atoms depend on both $n$ and $l$.$^\dagger$



The $n$-dependency was later successfully rationalised by the Bohr model, but saying that "the Rydberg formula only works for hydrogen-like atoms because the Bohr model only works for them" is misleading and misses the point, as:


This implies that the Rydberg formula was derived from the Bohr model, which is not true; it was merely empirically determined, and the formula predated the Bohr model by 25 years.
The Bohr model simply does not work for hydrogen-like atoms. The fact that it reproduces the Rydberg formula should merely be considered a serendipity; Bohr arrived at the correct result by the wrong method.
It does not lend any real insight into the proper reason why the Rydberg formula does not apply for helium, etc. (which I briefly mentioned above).




$^*$ In fact, the energy levels of hydrogen are not only dependent on $n$ (due to various small effects such as – but not limited to – spin-orbit coupling, and hyperfine splitting). Wikipedia has a good overview of the topic here and most QM textbooks have a chapter on the hydrogen atom, where they discuss these perturbations to the Hamiltonian and their effects on the energies. Not surprisingly, the inability to explain this was one of the failures of the Bohr model.

$^\dagger$ Of course, there is also a series of approximations here. The energy levels of multi-electron atoms are only approximately described by sums of orbital energies, so the transition energies are only approximately equal to a difference in energy between two orbitals.