Helix/Kraken reports a “banned user” as 404 not found, so you can’t tell the difference between (when looking up by name or ID) a ban, a delete, or a rename (when looking up by name).
TMI/chat will returned a suspended message on join attempt but that in itself doesn’t mean anything on it’s own (msg_channel_suspendedNOTICE Message IDs | Twitch Developers)
The Twitter account you refer to I don’t know how it does it.
So to my knowledge they are either poking something they shouldn’t or assuming that a Suspend TMI message means they are banned.