Yep. And so can any random citizen, as long as the First Amendment right to free speech is still in force. Of course, you're not required to respond by producing the ID unless the requester identifies him/herself as a LEO and provides proper evidence to back that up. And this off duty officer did let the request drop . . . until the poor oppressed RKBA-exerciser finished up his lunch and decided to reopen the matter by going over to the still-lunching off duty officer and demanding his ID. Judging from the whole story as it's presented here, I suspect that demand was made in a sufficiently aggressive manner that, combined with the openly carried weapon, made it at least within reason to arrest him and charge him with disorderly conduct.
The officer's main error, as far as I can tell from the obviously biased article, was in not being perfectly clear about what "hat" he was wearing when he initially asked "Who are you?". I'm still not sure if he intended the original request for verbal identification to be a formal request from a LEO, or a random question from a citizen which the requestee was free to decline to answer. The fact that he was lunching with a uniformed police officer, but was not in uniform himself, made his identity and role a bit unclear, and he was at fault for not making himself perfectly clear in that regard. The requestee couldn't really have known at the outset whether he was an off-duty or plain clothes LEO who had legal authority to demand identification, or a non-LEO friend of his LEO lunch companion who figured he could get away with making pushy demands like this because his friend is a LEO. This careless error on the part of the requesting LEO is what's likely to get the poor oppressed RKBA-exerciser a "not guilty" verdict when he gets his day in court.
That’s basically what I was thinking. The officer was remiss in not identifying himself as a LEO (and may also have shown questionable judgment on how do deal with legal open carriers). The other guy seemed to think that even if the other guy was a LEO, he needed RAS or PC to ask for ID, which I believe is not presently correct, plus he re-escalated the situation after the other guy was willing to let it go. Fault on both sides.