He may not have been under arrest but detained (a temporary state) but you can’t do that all day.
The officer's second problem is that there would have to be a record of her radio communication seeking to identify the subject if she knew that he had given her a false name (otherwise, how would she know?) Giving the officer false information (as opposed to just politely smiling) would provide the officer with reasonable grounds for continuing the stop,but claiming that the subject gave her a false name and that she majically knew it to be so would be weak under any circumstance, but is going to cost the government some real money in a false arrest civil suit in light of the presence of an unfriendly eye witness.