[XKCD has a blog called "What if" that has answered this very question](https://what-if.xkcd.com/150/
). I think they did a really good job. To quickly paraphrase the post, if there is more than one sun, you will have more than one rainbow! Depending on how close the suns appear in the sky, the rainbows may overlap!
If both suns were in roughly the same place you could see two rainbows (or four, y'know). To see a rainbow you must be looking into rainfall with the sun at your back. Two suns can't be at your back unless they're in the same parts of the sky.
A good rule is the following: A rainbow is light which is reflected^* at a wall of falling rain.
^* This reflection is not the direct reflection when the light first enters the water droplet, but the reflection at the back of the droplet, after the light has entered.
This means that with more light sources, you can see more rainbows.