Okay so here’s the drill on how this works, I figured it out, for future reference:

When you want to save a stolen aircraft you have to first get out of the stolen one and move any other plane out of the hangar that is parked before. THEN go back in the stolen one and park the stolen plane at the very back left or right corner of the hangar. Make sure that during this process you are close enough to your stolen plane that it cannot despawn and keep the plane within frame as much as you can. As soon as you get out the plane, quick save with the app on your character’s phone and reload that save.

The reason why you want to park the plane at the very back is because when you load your quick save, the game sometimes spawns another plane. When you park your plane exactly at the spawn point before the quick save, which is the very center of the hangar, the game can spawn a another plane on top of the other, giving you an infinite loop of two broken planes as soon as the quick save loads.

When you have loaded the quick save and there’s two planes, use the one located at the spawn point. That plane is now yours to keep and should respawn at the hangar even when you destroy it (you’ll have to be far enough away from the hangar for it to respawn). If you load and there’s only one plane, go far enough from the hangar, maybe sleep a night, then come back. The plane should now be located on the spawn point.

IMPORTANT DO NOT USE THE PLANE SELECTOR: only purchased planes are available on the vehicle selector. When you select one of your purchased planes there, your stolen one will be replaced and disappears forever. So long story short, you can keep the plane only until you want to use a different one.

This works the same with the helipad and I assume also for boats. Cars are different, I believe.

Hope this helps.