There is nothing stopping you in principle to keep learning on every interaction in an LLM. Practical benefits are likely small though.
notice that LLMs "context" chat is not learning, it's a temporary effect that gets lost as soon as the chat closes.
is it "learning" in a certain sense? sure, the same way like all indoctrinations are sold as "teaching". the model "learned" to be rapresentative but forgot what 1940 nazi soldier was like...
and fine tuning is not a scalable approach because it has human feeback in the loop. could they fix this error with more fine tuning? yes and they tried abut then the users simply asked "give me a picture a viking warriors" and the problems was again there, you can't fine-tune everything even if we assume the purpose is always noble.