A laser is built on quantum mechanics to create a beam of photons with the same frequency and phase. Someone told me a free electron laser is a based on classical electrodynamics. Is that true?
is classical laser possible?
IMO, the question doesn't make sense. "Classical" and "quantum" are not different options for how a thing can work. They are different options for us to try to understand how it works. LASERs aren't "built on" quantum mechanics, but rather, quantum mechanics is an appropriate tool for understanding stimulated emission. A free electron laser isn't built on classical electrodynamics, but classical electrodynamics offers a sufficient explanation for why it emits light.
The crux of regular laser’s working principle is the population inversion of the lasing medium. This is a quantum property arising due to the discreteness of energy levels.
But when you have a free electron, quantum mechanically the energy spectrum is now continuous. So the electron no longer has to jump discretely and continuous change in the energy (parametrised by $\vec k$) is possible. Since the wavefunction can now readily expressed as a plane wave, the classical treatment of an oscillating charge with harmonic time dependence gives the same result.
In the Wikipedia article
In most cases, the theory of classical electromagnetism adequately accounts for the behavior of free electron lasers. For sufficiently short wavelengths, quantum effects of electron recoil and shot noise may have to be considered.