I'd say it's a rare thing for King Charles to meet and greet non-royals, but that would be a total lie. The man is often out and about, shaking hands with and chatting with members of the public.
Take, for example, his royal tour of Australia and Samoa with wife Queen Camilla. The now-76-year-old monarch (yes, his birthday was yesterday and, yes, the family released a new portrait) made a number of public appearances during the tour where he met dozens and dozens of complete strangers. So, needless to say, the man is used to making small talk...which is why it's so surprising that he was just part of such an awkward interaction with A-list actor Denzel Washington.
The famous men met and shook hands on the red carpet at the London premiere of Gladiator 2 and, well, I'm sweating just thinking about how strangely it went.