It might seem obvious now that it had to be a frame, but on Inauguration Day there were few clues. Mrs. Trump delivered it to Mrs. Obama after she exited a limousine outside the White House and the couples shook hands and hugged. On a day full of pomp and protocol, the couples were then supposed to pose for a photograph, as two Marines saluted by their sides.

But instead of pivoting for the photograph, Mrs. Trump handed Mrs. Obama a rectangular box wrapped in robin-egg blue paper and a white bow. Mrs. Obama grabbed it with two hands and then panicked. It was not part of the script, she confirmed on the show.

“I mean, this is like a state visit, so they tell you that you’re going to do this, they’re going to stand here,” Mrs. Obama said in the interview with Ms. DeGeneres. “Never before do you get this gift, so I’m sort of like, O.K.,” she said.

Holding the box, she turned to her left toward one of the Marines, glancing in all directions for a spot to place the gift. The Marine did not budge. It could not go on the ground. At last, she appeared to give up, resigned that it would appear in an official photograph to commemorate the transfer of power.