You can come up with whatever mathematical definitions you want. When I say "It makes no sense," I'm speaking very precisely: There is nothing in nature which corresponds to the idea of multiplying two RGB colors together.
In that context, it makes no sense to multiply two colors together.
There is. There is energy attenuation in different frequencies. Look at an object of any color under artificial light then take it outside. Now the light hitting the object's surface has orders of magnitude more energy yet you still see it as same color. Because its surface attenuates the incoming light the same way and your brain detects that attenuation as "color". Speaking of which (the brain), the "color" does not exist in nature at all, it's just an effect of human brain analyzing image projected on the retina. So let's agree that there is nothing in nature which corresponds to the human concept of color.
In that context, it makes no sense to multiply two colors together.