In a short response, we can assume the answer to this question is NO, although science may never be able to completely know. Throughout hundreds of millions of years of evolution of the eye, humans have gained this ability to perceive color.
The concept of color as we know it is created solely in our own brains, an illusion if you will. The question highlighted here may forever be unanswerable, something scientists call PAP (permanent agnosticism in principle). Although there are ways to observe the differences in color perception, the question relates to two people identifying an object given the same initial conditions. No person will never be able to observe the experience of a color inside your mind. 
Let’s look at an example given by the great YouTube channel, Vsause. In the picture to the right, how do I know that when we look at a strawberry and in my brain, one perception exists, that the learned color in your brain is not something completely different (the one one the right). We have both learned to call this color red, we communicate effectively, and we walk away never knowing our differences.
The illusion of color is created in our brains. The brain breaks down certain wavelengths to give us the color spectrum. Humans have photoreceptor cells in the back of the eye that make color a reality (a diagram can be seen below). 
However, humans are not perfect. Color blindness is a condition in which the number of cones in one’s eyes is limited. In addition to colorblindness, there are many ways to see how colors are different to different people (remember the black and blue dress from last year?). It is important to understand that many with versions of colorblindness are not aware of their condition unless tested. But, we can test this because one may not be able to see something that should be noticeable to others.
Just as many birds can see well past our color spectrum and see into the infrared range, there are certain people that may be blessed with a version of superhuman vision. In a condition known as tetrachromacy, many women in the world today may be able to see millions more colors than the rest of us. The science on this issue is very young as we are just now understanding the genes behind the cause. It is believed that these lucky people have a mutation on one X chromosome, allowing them to have a cone which peaks between red and green colors, allowing for them to see extraordinary colors. Maybe you, yourself have this superhuman vision and do not know it. However, if you are a male, chances are you do not.
