Is there any objective, scientific way to prove that we all see colours the same
There are objective scientific tests which show that we don't all see colours the same, such as the Ishihara test for colour vision. Most people don't even see the same "colours" out of both eyes. For many people the left eye might see things more saturated than the right.
The question should also perhaps be refined a bit. Shouldn't it be formulated as whether we see things (objects, surfaces, volumes etc.) in the same colours? "Do we see colours the same?" as it stands seems to mean, "Do you see red as I see red?" But this presupposes that we are both seeing red, and then the question seems to ask whether we see it the same way, for example with the same degree of saturation or exactly as blue.