I've received reports of people purposefully impersonating users currently chatting, using their name and profile picture. The only way to fully tell them apart is to view their account ID in their profile page.
In addition to not allowing someone with the same name as the Host to chat, it may be a good idea to perform a "cosmetic" change to their profile name to include their ID for telling them apart from another user if they enter a Livestream chat and the name is already taken.
Bob (account ID #1234) is currently chatting. His chat is displaying "Bob" with his messages. A second user, also named Bob (account ID #5678) enters the same chat but when he talks, it displays his name as "Bob (5678)". No change is made to their account name (both would still show "Bob" when their profile page is viewed) but now it's clear that there are two users named "Bob" talking. If both says something, one would be from "Bob" and the other would be from "Bob (5678)".
If the first Bob refreshed the browser (or left & came back) and his profile name was still "Bob", he would be displayed as "Bob (1234)" if the second Bob was still considered live in chat. If both say something in this scenario, one would be from "Bob (1234)" and the other would be from "Bob (5678)".
Not a perfect solution but it would at least make it easy to tell separate people with the same name apart and discourage impersonators because it would be so easy to identify them as such.