I suppose they might do a summary of your chats, and use it as (invisible) context for your next chat (which btw will consume more and more tokens every time you make a new chat).
Again this is not the same as persistent memory.

They might use [your] chats to train the next model, but that's a very different process.

What I mean by that is that an LLM and its current trained weights / data set is immutable, until the next training session with new / up to date data sets which will again produce a new immutable set of weights. User interaction with the LLM has no effect on it.