I was thinking about recovering the actual bullet. Not any color mark left on the target paper. That never even occurred to me.
Three guys shoot at a deer. They're all using the same cartridge (notice the shotgun slugs?). The deer is hit one time. Who's slug hit the deer? If they each had slugs of a certain color you could tell pretty quick. In a lot of shotgun States group hunting is legal. I’ve been party to many a good nature argument over who should put their tag on a downed deer. “I got it. It’s blue. That’s Bob’s deer.” Not a lot of room for argument.
Two cops shoot at a bad guy. The bad guy is hit but so is a little old lady half way down the street. Who shot who? This would be an easy way to find out without having to wait days or weeks for CSI lab testing. “Officer Smith, good job. Officer Brown, we need to talk.”
Not advocating for colored bullets, or saying they're a good idea. Just spit-ballin’ what they could be used for, since someone asked. We all know bullets don't disappear or designate after you shoot them. Much of the time, especially with handguns, the bullet will be found inside the target, or directly behind it a few inches under the soil.