If you are focusing well, stars are comparatively likely not to occupy significantly more than a single pixel. But pixels are covered with a regular grid of color filters, the Bayer filter typically using an RGGB arrangement for 2×2 cell grids. A so-called demosaicing algorithm making use of redundancy/correlation of luminosity information then tries to reconstruct RGB information. But if a star lights only a single pixel, there is no redundancy/correlation to work with for estimating the color distribution.
So if you want a good estimate of colors, you'd need to defocus a bit so that stars get a chance to touch more than a single pixel. Systematically, you can do that by using diffraction, namely very small apertures. Ironically, it may also help if your sensor resolution is better than what your camera optics may be able to deliver.
You may also try recording raw images and then playing with various demosaicing algorithms: some may work better with the inherently problematic situation (possibly by leaning to a stronger default behavior of preferring to guess "white" in the absence of better information).
One thing also worth noting is that color filters have wide selectivity, and demosaicing algorithms tend to assume correlations between colors due to the image elements mostly being reflective and sharing a common illuminant. That assumption does not work on a star picture because every star has its own independent color spectrum. So this can be a reason that more complex demosaicing algorithms usually considered to be superior can actually work worse in this situation, making differences to the exact alignment to the pixel grid produce worse color variation than a different algorithm would.