My point is they would see those colors in the real world, but only in combination with other colors, so it wouldn't be that noticable. They wouldn't see pure versions of them much, if at all. The intensity in a dream might be much stronger, enough that it would seem like a qualitatively new color rather than "hmm, that blue has something different going on compared to other blues" they you might get in real life.

You could put them in front of light source that generated pure wavelengths designed to selectively activate their extra cone and ask what they see when you start mixing pure blue or green into it. Maybe they'd recognize those colors from their dreams, maybe not. I don't know. But it's a mechanism that would explain how tetrachromats might dream colors that trichromats don't.

But this assumes the gamut of trichromacy is more anchored in reality (woken life) than the gamut of tetrachromacy. In principle there is nothing different between a trichromat and a tetrachromat sense of color. They see one dimension of color more, just like humans see one dimension more than dogs.

The fourth cone in tetrachromats overlaps quite a lot with the other three (which is why most tetrachromats don't see more colors than trichromats). For people where the fourth cone is meaningfully distinguishable from the others, it doesn't seem weird to me that you wouldn't see many objects that are strongly activating just that extra cone and not the ones next to it (wavelength wise) because they're closer together.

We don't see bright, "pure" magenta very much, and when we do it's usually because we've engineered it. Nobody is engineering pigments for tetrachromats to enjoy.

If I'm right, people who are nearly (but not entirely) colorblind (anomalous trichromats) due to a shifted cone would likewise be able to dream in colors they are capable of seeing only dimly while awake, whereas dichromats wouldn't.