I did some testing and I think I know what's going on. If you tell your computer to remember you on TGT, it will automatically log you in when you come to the web site. If you're banned, however, the software will not show you anything except an error message, if we provided one, explaining to you why you were banned.14:28 Sep 15 2020 shows tello1 ONLINE NOW.
How can you be BANNED and be ONLINE?
However, each attempt to visit TGT, even if all you see is that error message, updates your user profile with a new date and time in the "last seen" field. Note that in that field it no longer says where the member was last seen. That's because the banned account isn't actually seeing any TGT content; they're just seeing the page headers and a message that reads: "Oops! We ran into some problems. You have been banned for the following reason: <insert ban explanation>."
I suppose if a banned member occasionally refreshed an open tab with that error message showing, they could continue to show as online forever. I'm not sure why they'd want to nor can I see where it hurts anything.
This idiosyncrasy doesn't bother me and unless someone can give me a compelling reason to do otherwise, I'm not going to suggest anybody up the food chain try to address it.