Hey there!
The HUD is actually a text box, that's forced to stay on screen via some edited code in the engine. You then store a lot of the information in variables that are updated every screen, the Room Names are manually written out and a switch is used to determine which room's text box appears, and Profile Pictures are built into the ascii file itself and typed out in the text as well!
It's using modded engine files that were originally developed by some users on the discord. If you're not a member, I recommend checking it out, there's a lot of cool work being done to push GB Studio's limits!