Just something to distinguish them better other than their name, as you might have multiple different copies of the same class to use different masteries. It’d let you figure out which character is which at a glance on the character select screen. Also means you’d be able to choose whatever name you want without choosing a name specifically for the build it’s using, like a lot of people do.
Might have to make room for it with the character name and level and stuff, but I think it would be very handy. A popup on hover could work, but I feel it should be visible without hovering over each character, even if it’s just one or two words.