The definition of an astronaut:
“The Fédération Aéronautique Internationale (FAI) Sporting Code for astronautics recognizes only flights that exceed the Kármán line, at an altitude of 100 kilometers (62 mi). In the United States, professional, military, and commercial astronauts who travel above an altitude of 80 kilometres (50 mi) are awarded astronaut wings.”
So yes, that does make him an astronaut.
Think bigger. Musk could probably walk into a meeting room with Mango Mussolini, strapped up, and he wouldn’t be searched.