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It appears that throughout your question you are touching on multiple questions and topics. I will address them in a series of quotes and responses, beginning with the title:

Are colors real?

They are not physical things. Colors are a form of perception (an abstraction). They exist in your head. In physics the perception of colors is caused by light waves; a form of radiation.

We know that we distinguish colors when they fall on our photo receptors in our eyes and neurons pass its signals to our brain to recognise them as the specific colors. How accurate is it?

Color perception is subjective. It is affected by all the matter that would affect the interpretation of colors - your eyes, nervous system, and brain. Accuracy would require a benchmark to weigh relativity from. How can you benchmark the difference between your perception of colors and something else's perception of color when there is no way to record how much somebody perceives color? You cannot say that Bill sees this much redness, that much green, and 5% less purple than normal without a strong benchmark.

...There might be more colors than primary colors can make up which our eyes cant interpret it as a different color itself.

Because colors are defined as combinations of light waves at certain frequencies, what you're suggesting is that there are more combinations of light waves than what humans can see - everything beyond the visible light spectrum.

My question is, is there any experiment which can tell us the color we see is the only and real color which is reflected by the object?

There is only one actual color (aka combination of light waves) from a certain object that you will see at any given time. If, however, you are asking if the color you perceive is the color you should perceive, then the answer is no. You can't without a very strong benchmark to base your observations upon and a method for determining the accuracy of color interpretation in a given organism (because it requires a way to measure all the components that make up or affect perception)