It accomplishes a whole lot more than you think. Say later the white woman decided to file a complaint because "the cops that responded didn't even care enough to take down her name!" or some such complaint (yes, there are such petty complaints that happen) against the force. Or later on the black lady files a harassment complaint against the department. There will be at least a written record of contact with said individual. Sure, the have the body cams and whatnot, but prudence demands at least having a name to have as a just in case measure in the days where people will file a lawsuit at the drop of a hat. They can go back, research when and where it happened, bring up the footage of the incident, make sure the department's ducks are in a row if something was to happen. But rest assured, most contact with the public in this kind of situation must be recorded on official documents for the safety of both the individuals and the department itself.
There was no reason whatsoever for said lady to get stupid about giving a name. "My name is Jane Smith." Simple, effective, done. She got completely crazy when she did not need to in any way, shape or form. As for arresting her, no, I think that's a bit far and there were certainly other ways of bringing down the situation if I was to Monday morning QB it. But then again, I don't know if it's a local or State law that requires you to identify yourself verbally to the police. Could be, I don't know and I'm not really in the mood to check.
And I'm going to ask you, however unlikely it is, to refrain from taking anything out of context here like you typically do.