I, too, assumed Prince Rainbow was a... "rabbitification" of mere rainbows. Seeing as they're probably uncommon enough for the rabbits to assume they're important messages from Frith, it makes sense. Also, when Pipkin sees the rainbow in The Mysterious Visitors, he explicitly says Prince Rainbow is visiting.
There is certainly a strong case for Prince Rainbow being a rabbit instead, however. The El-ahrairah stories have Prince Rainbow telling El-ahrairah to do things directly, and that can just as easily be interpreted to mean that Prince Rainbow is an actual entity rather than just a rainbow to the rabbits. Also, the case in the series I just mentioned could also mean that Prince Rainbow is supposed to be at the base of said rainbow rather than being the rainbow itself.
You know, just seeking out this information is really making me doubt myself. I can't tell which is more convincing anymore!