The ability to track separate conversations with like a side menu or with tabs would be rather useful in 2 different situations for us. It's similar reasoning to opening a new tab in a browser:
Depending on the user we have different Prompts and it's possible to have more than one available. As it is, to implement this we have to handle message history ourself so that switching between prompts doesn't send chat history from different prompts or we can create a custom component (this is actually what we do)
occasionally you don't necessarily want to lose the chat history but you don't want it affecting responses for other inputs either (similar to above but using the same prompt). Sometimes you want to preserve chat history to reference it later on and don't want to play the "scroll 50 miles to find something" game, so starting a new conversation is useful here also.