Wouldn't this be relatively easy if we wanted it? We could just feed it a "time step" token periodically?
Being good at generating questions, could become almost as important as answering them.
However, to make it more human-like, I would think they would make some type of AI self reflection. So like we can talk and think out problems to ourselves, perhaps they would have the AI periodically practice questioning itself, when humans are not around or asking it questions, and do some form of self-analysis.
A chatbot with a sense of time learned by also training with response rates, complete editing sessions, some formalization of interlocution, etc. would be interesting, but that's not what these are.