He isn't wrong. It would be like saying "I have read the Philosopher's Stone series". The series isn't named after the first title in it.
She got tripped over because that's what the TV show does.
Him: Name ten books
Her: Are you serious?
Him: Yes, 100%. Name ten books.
Her: Well I’ve read Game of Thrones. I’m obsessed with the twilight series.
Him: What’s the book called?
Her: Game of Thrones…. The series?
Him: What’s the book called, it’s not called Game of Thrones.
That’s literally wrong. There is a book in the series called Game of Thrones. Answering his question by saying game of thrones is an acceptable answer. Trying to do a gotcha over the series title is dumb if he’s himself technically wrong about there not being a book called game of thrones. Especially since he initially asked for book names not the names of series.
He would be equally as wrong if he asked for the name of a book and someone said The Three Body Problem and he was like “that’s not the name of a book, the series is called Remembrance of Earths Past.”