A cop can ask you your name for any reason he wants. Question is, are you lawfully required to answer. In most cases, the answer is no, you do not have to answer nor provide any ID (if you're driving a vehicle that's a different story). The only way a cop can force you to answer is if he has reasonable suspicion to believe you may have committed a crime. The deal is, the cops know exactly what constitutes "reasonable suspicion ". In the vast majority of cases, if a cop says he had reasonable suspicion , a judge is not going to disagree.
As far as calling for a check, a cop is never "off duty". He can always call and have a check ran.
Yes a on duty LEO can ask....and maybe you must comply. An OFF DUTY policeman can ask but as far as being compelled...nope. He might not even be in his jurisdiction. I really dont think an off duty cop can "call in" a name, that would mean they are detaining you.
Im still not buying into this story. Off duty and acting strange, doesnt warrent calling in a name and running a background check. Theres more to it. Somebody has an agenda. Or an axe to grind.