While the gift of seeing auras is a function of the third eye, it is actually a type of color synaesthesia.
For those who don't what that is, have you ever heard of people who see a certain color with different letters, or who can see what color music notes are? Those people have synaesthesia.
Seeing auras is referred to in the science community as "emotion color synaesthesia", but there isn't nearly as much information on it as other types. Sound color synaesthesia is the closest, and if you are truly confused on auras, you may want to read a bit about that.
People like this don't have a specific name - like psychics, empaths, clairvoyants, or any other type of specific name. The most common classifications are empath (those who feel emotions), or clairvoyant (those with "clear sight"). The term "empath" will be used in this site, but the information is the same for those of you who classify yourselves as something different.