With all the new Assistant features coming, and as discussed here #49165, I think we need an infrastucture to:

• save the preferences
• save the history (I would make it explicit as I'm not sure we want to save the history for everything)

We also probably need a button somewhere to clean up things.

I would do it per project.

For the preferences, I would probably use local storage.
For the history, maybe using something like IndexedDB makes sense. But maybe local storage is fine too. I have no idea of the constraints of these.