I was never able to find it myself. While writing I just guessed it was a gerbil or something. Glad to know you kinda found it and that this game was not just some fever dream.
[edit]
By the way, Internet Archive adds some sort of translation layer for flash animations so it can be opened in modern browsers including on your phone:
I’ve also heard it used almost exclusively in terms of “getting rid of someone/something from where it currently is”, most commonly “86 that thought/idea” in place of “forget that” or similar
It’s always been from restaurants as far as I remember. I’ve only ever heard it in restaurant contexts; i.e. “we’re 86 French fries” or something like that.