I've won arguments against ChatGPT where it told me that it would remember the info for next time, but ultimately reset to the models last state unless I repeated the conversation
While it might present a save menu similar to the real game, that doesn't mean the menu itself actually functions. With LLMs, they are ultimately only able to remember what's been pre-trained into their model + whatever is discussed within their context window.
A hypothetical LLM based online bot/shill sent out into social media would likely be including the entire discussion within their context window for each post generated though, otherwise it wouldnt really be possible for it to maintain a coherent conversation.