Is the source of light from flame tests the same as that from "neon" signs?

Yes, the source of light is the same in both flame tests and neon signs because the phenomenon producing the light is the same.
The phenomenon is the emission of photons by the de-excitation of electrons. The electrons present receive energy and transition to an excited state, which has more energy than their ground state. When they transition back to their ground states, they emit their energy in the form of photons, which are seen as the colors produced.