Actually a rainbow contains the entire visible light spectrum. Our words and terms for individual colors is (somewhat) arbitrary.
There is nothing inherent in physics of color or our eyes why, for instance, light with a wavelength around 470 nanometers should be a distinct color (in this case blue). It seems like a strange idea, because our use of language cements the notion that these colors are somehow special when compared to light at intermediate wavelengths.
Tests have been done with children of different cultures to see if there is some hardwired circuitry in the human brain to identify distinct colors the same way. In other words, we see red, blue, green, but does another culture see (what we call) purple, blue-green, and yellow-orange as their primary colors?
As an aside it is interesting to note that ancient Greek and Roman texts, for instance, tend to have a lot fewer color terms than modern works of literature.
An early test in the 19th century where anthropologists did the ask-kids-color test suggested that in other cultures people tend to pick out colors from the base set of 11 in English (red, yellow, green, blue, brown, purple, pink, orange, black, white, and grey) - suggesting that there is something to the idea that these colors are hardwired into the brain.
On the other hand, at least one more recent study found the opposite - that when shown a selection of hues children from another culture did not pick out the 11 English terms as special or primary.
(Later of course the system gets refined, as physics lets us know how to combine colors in different situations. That is, once you pick "blue" as one of your special colors, then physics demands that the other primary colors are red and yellow. But it is not unique. If you start with purple then blue-green and yellow-orange are the other primary colors.)
But, regardless of whether or not some colors are hardwired into the brain, a rainbow shows a continuous spectrum across the visible light range and beyond - extending slightly into the infrared and ultraviolet regions.
So, the reason why a rainbow has the 7 primary colors is that these are the ones which we have identified and named as being "special."