A common problem when people with different backgrounds discuss primary colors is that they associate different notions with the term
"primary color". There are at least four different concepts regarding
"primary color":
1. a set of real colorants or colored lights that can be combined to produce a gamut of colors -> "real" primaries like RGB or CMYK
2. a set of imaginary colors that can be combined to produce the whole gamut of colors a human with normal vision can see -> color space primaries like XYZ or LMS
3. a set of colors that an observer perceives as pure -> psychological primaries or more precisely unique hues like those used in NCS or CIECAM02
4. the colors that are best examples of the (considered to be universal) basic color terms -> psychological primaries or more precisely focal colors (focal colors are a different concept than unique hues but empirically they seem to coincide with the unique hues, so the primaries from NCS or CIECAM02 are good examples here too.)
> (My vague and possibly-incorrect understanding is that any three hues can be used as "primary hues", but different choices will result in different hue spaces, and it's just that CMY and RBY produce spaces that are useful in printing and painting, respectively.)
If you are happy spanning any arbitrary gamut than any three hues will do. If your ambition is to cover the whole gamut of normal human vision no three real colors will do. You will either have to use more than three or allow imaginary colors. On the other hand there are infinitely many sets of primaries to span a certain gamut. Wikipedia expresses this probably much better than I can[2]:
> All sets of real and color-space primaries are arbitrary, in the sense that there is no one set of primaries that can be considered the canonical set. Primary pigments or light sources selected for a given application on the basis of subjective preferences as well as practical factors such as cost, stability, availability etc. Color-space primaries can be subjected to meaningful one-to-one transformations so that the transformed space is still complete and each color is specified with a unique sum.
[1] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1404500/
[2] https://en.wikipedia.org/wiki/Primary_color