As the camera cannot really see color, there is an array of color filters in front of the sensor that detect basically only brightness (for that color due to the filter). The most relevant pattern for these color filters are called a Bayer matrix. Some camera manufacturers use a different pattern, namely e.g. Fuji with their X-trans Sensors.
This means that the RAW file processing does something called demosaicing. Basically several hardware pixels or photo sites in the sensor are grouped to form an image pixel. In a Bayers matrix, this is one red, one blue & two green ones. The three colors red, green and blue (RGB) are then used to determine that pixels color.
If there is much noise in the image - which means randomized jitter of the correct brightness value, it can lead to the colors becoming pretty unstable. If the three color jitter values are e.g. less red, less blue, more green, then you will end up with a pretty pronounced green cast of that pixel's color.
So usually the brightness levels of color noise is less severe as it is evened out a bit by the grouping of the hardware pixels, but the color shift can be pretty severe.
So: The cause is really the same: Signal to noise ratio becoming worse. However the effects in the final image are different.