The underlying LLMs do, but we choose not to use the capability because it's expensive and doesn't quite work as well as we'd like it to, or quite in the way that we'd like it to.

We are perfectly capable of running LLMs in a way that does a backward pass to update some or all of its weights after every user message. But, naively implemented, you only get partial, fragmentary absorption of the info in those messages, it costs three times as much compute, and you lose out on the ability to implement a ton of optimizations that making modern LLM serving economical.

If you want to do it, though, ask your friendly neighborhood robot to get it working with a tiny model (whose full precision weights fit several-times-over on your machine's resources).