“Great news! Tonight, we broke the all-time attendance record previously held by Elton John at #SNHUArena in Manchester, New Hampshire!” Mr. Trump tweeted.

In the days that followed, Mr. Trump repeatedly asked aides if his victory over Mr. John was capturing headlines. It wasn’t. Though aides were not surprised as they did not view the accomplishment as a major story, to the president, it represented an emotional wound — his belief that he is perpetually demeaned and never receives his due, according to people close to him.

It was not the first time that Mr. Trump focused on topping the attendance records of Mr. John, who was knighted by Queen Elizabeth II in 1998.

In Montana last summer, Mr. Trump asserted he had broken Mr. John’s record at the arena where he appeared — and again complained that he was not getting credit.