Umm no, how is asking "where are you from" and then badgering then to divulge their ethnic origin the same as asking someone which university they went to? That part of the conversation should have ended when she said she was from London.
I have no idea if she's exaggerating what happened, but it's unacceptable to be harangued into answering where you are from, if the person asking won't accept your first answer.
I'm racially ambiguous, not white, have very few links to my country of racial origin, am British born and raised, identify as British, and always dread the 'where are you from' question. A white person with a British accent wouldn't be asked this question, so why am I? It invalidates my entire upbringing and history, and makes me feel like an alien and not accepted.
I do give a pass to elderly people asking me this question, however. And I wouldn't go running to Twitter if this had happened to me. Ngozi clearly has an agenda, but that doesn't mean what was said to her (if it happened) was right, or not that bad.