Perhaps a nitpick, but 256^3 doesn't really represent anything about human perception-- it's just how we store colors in computer memory. The human eye can perceive a far wider range of colors than any screen can (currently) represent. It's just as valid (and probably more accurate) to use floating point to store color (rather than bytes), and in many high-fidelity applications this is actually how it's done.
The bit about colors "looking different" in different contexts also isn't really an "extra axis". The example you give is just your brain (attempting to) separate out the illumination from the scene colors. The range of actual colors you can perceive isn't increased by this.
Fundamentally human color perception is three-dimensional, because humans have three classes of color receptors on their retinas. There are also rods in the retina (the cells that are color-insensitive and detect only brightness), and in principle they could inform color perception, but this does not seem to be the case-- the intensity of light only shifts things around in the original three-dimensional space.
There are also a very, very few people (women only) with genetic mutations that give them four different color receptors in their retina. However, there is no evidence that their brains can take advantage of this extra information and allow them to perceive an extra "axis" of color. It is much more likely that the "mutant" receptors are lumped in with the stimulus of one of the others, which will shift their perception of color, but not expand it.
In short: There are three axes of color perception. Optical illusions don't change that.
The bit about colors "looking different" in different contexts also isn't really an "extra axis". The example you give is just your brain (attempting to) separate out the illumination from the scene colors. The range of actual colors you can perceive isn't increased by this.
Fundamentally human color perception is three-dimensional, because humans have three classes of color receptors on their retinas. There are also rods in the retina (the cells that are color-insensitive and detect only brightness), and in principle they could inform color perception, but this does not seem to be the case-- the intensity of light only shifts things around in the original three-dimensional space.
There are also a very, very few people (women only) with genetic mutations that give them four different color receptors in their retina. However, there is no evidence that their brains can take advantage of this extra information and allow them to perceive an extra "axis" of color. It is much more likely that the "mutant" receptors are lumped in with the stimulus of one of the others, which will shift their perception of color, but not expand it.
In short: There are three axes of color perception. Optical illusions don't change that.