the red background is probably reflecting red light onto the griffin, while the blue background is reflecting blue light. This will cause her to look different on either background. This is also the reason why Windstone never uses colored backgrounds in our product photos. We only use neutral colors, like black and grey.
Your camera could also be auto correcting differently for the different pictures, causing the griffin to photograph with slightly different colors based on the background. I know that with my own camera, if it sees mostly yellow, it will sometimes auto-correct towards bluer tones and vica versa.
Additionally, the human eye will perceive a color differently based on the background, even if the color of the main object never actually changes.