I am sure we can say "we checked into the hotel" because we want to use a room/service in the hotel.
We often say "we checked in at the airport".
I don't think "we checked into the airport" because we don't use any service in the airport.
If we are forced to say "checked into", then I think we can say "he checked into his flight at the airport". But I am not sure if we can say that.
Note: there is a difference between "check in" and "go through customs and passport control". You check in first before you go through customs and passport control.
Another note: we can not say "he checked out of the airport" but "we got out of the airport through customs"
Is it correct to say "he checked into the airport"?