I've never heard that rainbows are supposed to have a discrete number of colors.

It's always been the standard abstract that I've heard to teach around it.

In Ireland/UK it's something that gets brought up in primary and secondary schools, first when talking about rainbows and then later in discussions on optics/refraction — e.g. you'd get kids to make their own Newton's disc (https://en.wikipedia.org/wiki/Newton_disc) and see colours blending together to white.

I was taught it as "Roy G. Biv". I still remember what the letters stand for, but I don't remember the difference between indigo and violet.

That's because there isn't a meaningful distinction between indigo (as a shade) and violet. IIRC, Isaac Newton thought it was important (for biblical numerology reasons) that there be 7 colors.

German has 6, french has 7 though