The twitter account published the location of the plane using publicly available information. It could have also published the information of one of Virgin Atlantic's plane.
In fact it's not even Musk's plane, he just owns the company that owns the plane, Falcon Landing LLC. The twitter account didn't actually know if it was Elon Musk being flown or a bunch of burros being ferried back and forth between US states.
>FAA didn't create a plane tracking system so that a kid can put this info on a map in real time.
You're right, the FAA made the system open so that anybody could do this, not just this kid.