EDIT: Alrighty...this is [SOLVED].
I'll explain in my next comment.
OS: Linux Mint 21 x86_64
Kernel: 5.15.0-67-generic
Shell: bash 5.1.16
Resolution:
1920x1200 <- This is the monitor the game runs on
1680x1050 <- This is my extended desktop
DE: Cinnamon 5.4.12
WM: Mutter
CPU: AMD Ryzen 7 5800X (16) @ 3.800GHz
GPU: AMD ATI Radeon RX 5600 OEM/5600 XT / 5700/5700 XT
Memory: 26606MiB / 64244MiB
I have a ton of programs running in the background but I doubt that is the issue. I will try after a fresh reboot and get back to you.
I've not had issues with other Unity games afaik, but I will be paying attention to that in the future. The fullscreen button in the webapp version above works as intended. I'm running Firefox 110.0.1 (64-bit). Likewise it works in Chromium 111.0.5563.64. Of course the 'quit' button doesn't appear in that webapp version so cannot test it.
Oh my goodness - I think you cracked the case.
When you asked about my resolution I was like, "Hmmm what will happen if I attempt to run the game on my extended desktop (1680x1050)?" I set that monitor to my primary screen and ran the game. It works! Works great, exactly as intended - BOTH 'Quit' and 'Fullscreen' buttons work! So I think, "It's gotta be my (1920x1200) monitor's fault." I switch back to test again on the troubled monitor! And it works fine! :: visible confusion :: I just want to apologize. Definitely a workstation issue. Not sure which configuration settings change when swapping primary monitors but that was the key. Just switching back and forth fixed it. Sorry for sending us on a wild goose chase!