FYI This a slight variation on a fairly old board game called Can’t Stop.
In 1992 it would have more likely been a telnet service.
This is also very achievable in hypercard or visual basic so it could have also been a downloadable application
I've got a Next and an HPUX so I've tried the early browsers and written code for them. (HPUX is for ViolaWWW) They're pretty buggy. Making them handle the game would take some serious work