There is a way to do this in Vanilla Minecraft, using the Scoreboard feature. You don't even need to create an actual score to track. Just use commands along the lines of
/scoreboard teams add greenteam
/scoreboard teams add redteam
/scoreboard teams add blueteam
/scoreboard teams option greenteam color green
/scoreboard teams option redteam color red
/scoreboard teams option blueteam color blue
This example creates three rather unimaginatively-named teams, and gives each team its own color. Feel free to choose the names of the teams however you want. Then use
/scoreboard teams join <team> <player>
to put a given person on a given team. This will make their name take on the team's color in chat, above their head, and on the Tab menu of all currently-signed-in players.
While the team mechanics can be used to do things like team-based disabling of friendly fire and the like, the default settings are for a player's team to have no direct, actual effects on gameplay.