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For most people, the experience of color is most likely pretty similar. At the same time, there is clearly variation between individuals.
Color perception is a favorite topic of philosophers who like to ask: What if your color spectrum was completely inverted from someone else's? Would you be able to tell?
The answer from cognitive psychologists is yes, you would probably be able to tell. The reason is that from the perspective of how the brain is wired, the colors are not arbitrary. There is an internal structure to the color space that results from the relationship between the three color photoreceptor types and visible light, as well as the remapping of those photoreceptors to two parallel opponent color channels for red-green and blue-yellow and the relationship of color to luminosity and shadow.
It is certainly the case that people with red-green color blindness, who are missing one of the three color photoreceptors, see colors differently. This is often not immediately obvious, but they eventually figure it out. As one person described the experience: "There is this color, and sometimes people call it red, and sometimes they call it brown, and sometimes they call it orange."
So, how different is color experience across people?
Well, we have some clues. Even among people who do not have color blindness, colors are highly salient to some people and less so to others. Some people have an easy, almost automatic intuition for matching colors that go together in outfits, and other people have "no sense of color". In synesthesia, people can see color as intrinsically connected with letters, words, shapes, sounds, or personalities. But even for people without synesthesia, these associations are different from person to person and often somewhat stable. People have different favorite colors, and different associations between color and mood. Given all of this, it is safe to assume that people experience color differently.
However people also experience color in similar ways. Some of that is culturally induced. A white wedding dress is the color of innocence the West, but in China, a wedding dresses are bright red. Some color associations are biologically induced by the way the color system is wired in the brain. Some of it is undoubtedly learned by the brain's highly adaptive visual system according to the frequency of color association with different types of objects and situations in the environment.
The World Color Survey of the 1970s examined names and hue identities of "pure colors" across world cultures and found broad similarities with culture-specific differences. For example, all cultures have warm and cool colors, but not all cultures recognize "pink". 
In conclusion, subjective color perception across people is widely the same, but it is also different. Not the most satisfying answer!
 Philippona DL & O'Regan JK (2006). Color naming, unique hues, and hue cancellation predicted from singularities in reflection properties. Visual Neuroscience.
 Bompass A and O'Regan JK (2006). More evidence for sensorimotor adaptation in color perception. Journal of Vision.
 O'Regan JK, Noe A (2001). A sensorimotor account of vision and visual consciousness. The Behavioral and Brain Sciences.
 Berlin & Kay (1970s, 1980s). The World Color Survey.