It may have something to do with the resonance frequency of the phaser beam.
In The Best of Both Worlds, Shelby orders Data to alter the frequency of the Enterprise phasers during their first battle with the Borg cube.
SHELBY: Data, fluctuate phaser resonance frequencies. Random settings. Keep them changing. Don't give them time to adapt.
When we see the result on-screen, the beam color does seem to fluctuate until they finally break free.
Also, from the Star Trek: Deep Space Nine Technical Manual:
When phasers are fired by a ship with deflector shields active, the beam is frequency locked to the second-order harmonics of the shield emissions. This prevents the beam impacting on the shields and overloading them, or rebounding back at the firing ship.
This could explain why different space-faring races might have different color beams - it's reasonable to assume that each type of shield technology would operate on a different frequency range, so phasers (and possibly disruptors) would have to operate on a corresponding frequency to match.