I still remember when I first encountered this mind-blowing question. I was in elementary school, and it stunned me.
"How do you know we see the same colors? Maybe when I see red, you see what would look like green to me. But since we were both taught to call what we see then "red", we'd never know if we saw it entirely differently!"
An important step in one's philosophical development, in understanding the relationship between name and thing it names, and the vagaries of perception. But ultimately, when you get right down to it, it doesn't really make sense.
What would it mean to say that you see what would be green to me when you see red, on a physical level? When you see red, certain parts of your eye, your optic nerve, and your brain all have particular electrochemical reactions. If those same electrochemical reactions were to happen in my eye, optic nerve, and brain in the same places...
Wait, same places? My eye, optic nerve, and brain aren't even wired the same as yours. The neurons are in entirely unrelated places. One of us probably has more rods than the other, one more neurons in the optic nerve. They don't have analogous functions on the individual scale, only collectively.
It's like saying, if you took the electrical state of the photoreceptor in a Samsung digital camera and put it onto a Kodak digital camera, would it produce the same image? Well, the two cameras probably have a different number of components which are connected in different ways and serve different purposes, so how could you even figure out which bit of the electrical state in one to express in which way in the other? And even if you did, not only wouldn't it likely be the same image, it probably wouldn't even be an image at all. The Kodak would probably short-circuit from having voltages in the wrong places.
The sequence of steps that goes from "photons of a particular wavelength coming at you", through all the intermediate steps in the eye, optic nerve, and brain, and that ends with the concept "red", involves a huge number of arbitrary mappings of one thing into another, each of which is unique to each individual's particular biology. In fact, it's not even consistent for a single person. The electrochemical signals in your head that mean "red" today are not the same ones that will mean "red" next year.