In the lobby, it would show their names. And the client shows the names of people in a lobby even if you are out of the lobby.
To achieve this for a tournament, the admins would probably need to create new accounts to be used by tournament players. E.g. they could tell Tagada he needs to log out of his main account and log in to an account called "TournamentPlayerOne", and give him the password to that, and do that for 16 different players who all would need to log in that way.
Also the admins would have to find a way to hide the country flags in the client for those accounts. (Probably the easiest way would be to just shut down national flags for everyone for the day of the tournament--this is the way I think, lazy AF. That's easier than re-writing the code to specifically not show flags for just the 16 tournament accounts, although that also would be a good option, especially because the accounts could be re-used for later tournaments.)
The challonge would have to be set up according to their fake names, not the real ones, or we could figure out who they are just by looking at the brackets.
Probably the admins would have to assign each player a color so we wouldn't guess them by their color.
We wouldn't be able to tell them by their chat messages because they could intentionally do chat messages in the style of another Playerino.
Then it would be up to the TD to decide when each player's identity is revealed. (When they are eliminated? Or wait until the end of the tournament?)
Players would probably have to be locked out of their main accounts for the day of the tournament to prevent a defeated player from logging back in under their real account during the tournament, which would give it away.
Even if you created a parallel "game mode" like Galactic War, Aeolus would still show who was present in chat, so it would provide clues about people's identities. I think "tournament accounts" are the only way to make this work.