There are separately colors which we are wired to hypothetically receive, as you point out (and I find interesting) - IE, if you replace the eye with some sort of artificial replacement, then such a signal could get received by the brain, but with the current sensors it is impossible (also, I suppose a colorblind person could receive such a signal as 100% red without the other colors).
There's also the possibility during a dream of creating objects or sensations that are not physically possible. For instance, I've had dreams where I had a form of telekinesis, and obviously there's no sort of brain wiring to manipulate physical objects outside of the normal motor functions. Still, it made sense in the dream - and it's not like I could describe how I did it, and it's not because the hardware exists but the sensor doesn't.
Although - you could argue that there's a layer of abstraction in the brain of such things in the brain - for instance, when I'm typing these words I don't actually ever have any conscious awareness of my finger motions. I don't even think about the act of typing at all; only the words. So, thinking about it, the brain does have an idea of manipulating objects outside of a need for muscles to do it.
I would think for a tetrachromat the abstract sensation of seeing colors you can't describe to others is itself a familiar one, and that feeling itself is something you might encounter in dreams. My dreams tend to start with a feeling, and then build a context for it.
Tl;dr - dreams are weird.
The trick to impossible colors is recognizing that our existing receptor spectra overlap. It's not possible to get "100% red" in the sense of only the "red" cone cells firing and none of the "blue" or "green" ones, because those other cells are going to fire at some small rate for the same frequencies.
But our brain can certainly receive such a color even if our eyes don't produce it, so in some sense we "understand" it as a color even if we'll never see it outside a laboratory or hallucination.