I think it’s the other way around. I’m not an expert, but from my readings on mythology and languages, it seems that deities were often named as the noun of which they were the god of.
Thus, if I were to create my own deities, I might call the god of the ocean by the name Ocean and the god of the clouds Cloud and the goddess of the rainbow… well, I’d name her Rainbow.
If this is correct, then it certainly shows a different perspective on these mythologies, when you don’t have a separate name for them which is just a meaningless combination of syllables. When you have to say things like “Cloud caused rain to fall on me for a while and after he was done, Rainbow showed up.” This shows just how the ancients were really just anthropomorphizing natural forces when creating their deities.
But as I said, that’s a non-expert’s theory.
If true, then that explains the whole Iris thing. It’s not that Iris came to mean rainbow, in association with the deity of the rainbow. Instead, the word for rainbow (Iris) was used for the name of the deity of the rainbow.