The text prompt in the game might also be vulnerable to arbitrary code injection, but that wouldn't really have anything to do with the prompt injection being used here. Everything being done is within the confines of chatGPT which wouldn't need or have access to any of the game's code.
Which, now given the ability to inject arbitrary code, you could conceivably now write code to list every variable it had access to.
The text prompt in the game might also be vulnerable to arbitrary code injection, but that wouldn't really have anything to do with the prompt injection being used here. Everything being done is within the confines of chatGPT which wouldn't need or have access to any of the game's code.