It’s an aeroplane or an airplane but not a plane.
Plane as a shortcut to saying airplane comes from the fact that a wing is fundamentally an inclined plane, not a wood plane....I say plane all the time
In the very early days of flight they didn't say "to fly", the phrase used was "to plane", pity the easily searchable early copies of "Flight" are no longer available so you could see the use of the word "plane" in context. Then the US Navy has a person called a "Plane Captain", not involved in carpentry, but responsible for the maintenance of a particular aircraft, the airframe "belongs" to him, and aircrew get to have a lend of same.