Think of the belts and the colors as symbols. Humans gravitate toward such symbols as a "Symbol of belonging." That belonging is connected to human survival in a tribe or clan. Although modern times would indicate a loss of that need nature has not. Look to how humans tend to gravitate toward groups of all kinds just like karate-ka tend to gravitate toward a style or system that seems to match their personalities. This is why associations tend to be there along with such affiliations toward a dojo, style or system. We humans have a long way to go to rid ourselves of the trappings of ego, pride and survival as you find in tribes or clans. One of the first things many new tribes do is associate themselves with some symbolic representation that makes that tribe or clan "Unique," and so on.
Also, within any tribe or clan survival depends heavily on maintaining a hierarchy where everyone has a role to play that is inter-connected with that tribe or clan - for survival. Tribal members will look up within that hierarchy to increase their status and importance thereby a belt system in the dojo fundamentally does the same for that dojo's members.