The man, 47, was assessed by medical staff and again by a mental health team after the incident on Tuesday.
An RAF Typhoon was seen escorting the Qatar Airways flight QR23 from Doha after the pilot was handed a note about a possible device on board.
The man was sectioned "for a fuller assessment", police said.
The Airbus A330 was carrying 269 passengers and 13 Qatar Airways crew.
It landed ahead of its scheduled arrival time of 13:15 BST, the airline said.
Manchester Airport was temporarily closed and some other flights were diverted to Liverpool and Leeds.
Operations at the airport resumed at about 14:00 after flights in and out were suspended for about 25 minutes.
Passengers disembarked the plane "as normal", a Manchester Airport spokesman said.
The RAF confirmed Typhoon aircraft were launched from RAF Coningsby in Lincolnshire as part of its "quick reaction alert role" when a pilot requested assistance.
Aurang Zeb, 60, who was returning from a holiday to his home in Bradford, said: "I thought there was something wrong because there was a jet flying so near."
He said the plane landed and was taken to an area well away from the terminal, where they waited for 45 minutes, then moved again closer to the terminal.
Mr Zeb added: "Then I saw all the police with guns. Lots of police everywhere. Kids were crying, some people looked very worried because of rumours there was a bomb on the plane."
Speaking on BBC Breakfast, Matt Cox said he was asleep in his seat until the aircraft landed.
He said: "We didn't get told anything about the nature of what was happening at any point.
"It's not nice finding things out from Twitter rather than the people who are supposed to be in charge of you. Obviously we know why they did it."
A man has been sectioned under the Mental Health Act after a bomb hoax resulted in an RAF jet escorting a passenger plane to Manchester Airport.