It's a rather well-established fact that males have a higher risk of color deficiencies: About 8 percent of males and about 0.6 percent of females are affected by red-green color blindness. Most of these color deficiencies, in particular red-green blindness, are due to an inherited variations that are associated with the X chromosome. Thus, even though the phenomenon is more complicated than this simple account, at least the first part of what is stated in the video is true: information about light-sensitive visual pigments is stored in the X chromosome, and due to basic genetics, women are less likely to be affected by mutations of the pertaining sequences.
So, it's not wrongwrong to claim that "[this] is why females can distinguish more colors and shades", but this claim is only true in a statistical sense. On – on an individual level, it doesn't follow that an average woman will be able to distinguish more colors than an average man if both are unaffected by any genetically-caused color blindness.
There seem to be only few studies that have investigated gender differences in color sensitivity with unimpaired subjects. One somewhat recent study reports some differences between males and females with regard to the perception of monochromatic light across the visual spectrum. OneTheir first result concerns the translation of wavelengths to perceptual sensations:
The study supports the notion that males and females do not perceive colors in exactly the same way, and that there may be a range in the visual spectrum where females have a better discriminatory ability than males. Yet, the authors acknowledge that both effects are relatively small ("There were relatively small but clear and significant, differences[…"differences[…]"), and it's unclear how the findings, which were obtained explicitly usein a highly experimental setting, translate to everydayevery-day color discrimination.