You ought to give them your evidence against them. If they refuse, you could give them a light temp-ban. I doubt they'll do it again. Some rare cases, of course, could be an exception.
2 people already found out who I was before this.
Users are free to create a new account or switch to an old one, so long as they both inform moderators to ban the prior, and demonstrate no exploitative intent.
Users are free to transition a new account or back to a former, so long as they demonstrate no exploitative intent, and inform moderators to ensure only one is active.