Yes, there are other minds, and they see the same colours as yours.
There's a trick from physics. If an electron can't apparently know where itself is, but every other particle knows where it is, then the electron must know where it is, even if we can't figure out how. It works because the next position of the electron must be consistent with the observations of all observers, and to reach there, the electron had to have been in the observed location.
By yourself, the other-minds problem is apparently insoluble. However, the problem would only occur in the first place to a conscious entity, as there is no observable explanandum for unconscious observers. (Go on, start listing things that you haven't thought to explain because they don't exist.)
As per my last post, objective entities cannot properly pretend to be conscious. An unconscious entity can only parrot the words of the other-minds problem. (Wikipedia isn't conscious, but you can read it there.) Which means when you observe a statement of the other-minds problem, you can be certain there's a mind in the statement's past light cone. Combined with your certainty you have a mind, you can be certain there are other minds.
Relative to that other mind, you might be parroting, like Wikipedia. However, literally all other conscious observers can be certain there are other minds. When a fact is knowable from all perspectives but one, it is knowable from all perspectives. Even if you can't work out how.
If nothing else, I can back off slightly from pure idealism and note that it must have been independently stated as a problem by at least one other person.
I'm pretty sure but can't quite prove that conscious entities are absolute. If red is unmistakable, then when you think you see red, there's only one possible entity you can be seeing, red itself. (Plato was almost right.) Similarly, for the brain to talk to consciousness, it would either have to tune the signals after each birth (and tune it to a standard according to what? How would the standard know to be different?) or the signals are themselves absolute, so they evolve once and stay good.
Regardless, there's an entirely separate line of evidence that they're absolute, at least inside each mind. Conscious entities can be similar or different from each other, just like objective entities. Blue is similar to cold, and red is similar to warm. (Sharp sounds are similar to sharp surfaces, and smooth sounds are similar to smooth surfaces.)
We know from other complex phenomena that simply inverting them cannot preserve all symmetries. The opposite of red isn't blue, it's cyan.
Cyan is similar to energetic and blue is similar to calm - they are similar to each other in some ways but different in others, which means even if you also inverted calm and active, the colour characteristics would no longer hold. The red-warm-energetic relationship converts to cyan-warm-energetic - that fire still looks as warm and frenetic as it feels - but cyan was originally energetic, it should register as calm on the inverted scale. If you don't invert that scale, blue-cold-calm can't convert to yellow-cold-calm, because yellow isn't calm - someone who saw this way wouldn't find ocean waves soothing, because of the colour.
Meanwhile, warm and cold are direct opposites, but the sensations indicating such are not. Especially hot, which is qualitatively different from warm, as reflected by the fact that hot results from the warm and cold detecting nerves being excited simultaneously. You cannot change the relationships to make red similar to cold without changing the relationship between cold and warm and hot, thus changing the second-order relationship between the red-blue relationship and the cold-warm relationship.
A person with an inverted or rearranged colour scale would have different relations of similarities between their sensations, no matter how cleverly rearranged. Since we do not observe different relations or relations-between-relations, we can safely conclude humans see the same colours the same way.