Some modes are discrete, like electronic levels or vibration modes.
Some are continuous, like the translation of a molecule in an unlimited volume.
And some modes would be discrete BUT are so plentiful that we can't separate them, like translations in a limited but macroscopic volume.
One other reason is when modes have a short duration. Typically rotation modes, which correspond to faint frequency differences (high GHz, THz, far IR). Shocks between molecules stop the previous rotation mode and begin a new one. To observe the modes separately, you need enough observation time (Heisenberg or Fourier, depending on your background) so your frequency selectivity is better than the frequency separation of the modes. If not, too broad and close frequencies overlap, resulting as a continuum for the observer.
So you can observe rotations in a very low-pressure gas, for instance in radio-astronomy. In a liquid, shocks happen too often, and you get a continuum.
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With these overlapping modes, you get a continuous temperature scale in a first way.
But there is more. Temperature also tells the probability to get a molecule in some state (Boltzmann and the others), so even states are well separated (vibration of O2 at RT) a statistics over many molecules gives a temperature. 10.001% in the excited state is warmer than 10.000%.
This is not just a matter of definition and measurement. If you put to pieces of matter in contact and allow them to get in thermal equilibrium, one with well-separated states will really get its statistics to match the other's temperature.