No he hasn't. What Boxing organization does he fight under?
Yes, he has.
The Texas Department of Licensing and Regulation (TDLR) sanctioned his fight, on a WBC card. And Marvin just corrected me (accurately) it was not technically an exhibition (even though it was billed that way), that was the most legitimate fight of Paul's (novel, but advancing) career.
Granted it's an unorthodox way to drive into pro boxing circuits, but that's not unusual. I know guys who came from nothing and trained around garbage, and just started throwing down and now they are prize fighters in early retirement.
The "Cinderella Man" is a thing, it doesn't apply to Jake "Disney Problem Child" Paul but it inspires and that's important.
Right now Jake Paul is just a fighter who boxe
Right now he's suspended by the TDLR, along with Tyson, because that's standard for a professional boxer after a sanctioned, legally licensed match, as you pointed out. But even there, so many people are claiming he's suspended for some infraction. The spectators strike again.
Jake Paul is very likely on his way to becoming a pro heavyweight champion. He's on a winning streak against better and better fighters, and Tyson made him even more famous. And I really hope it helps him grow up a bit, but you know. Maybe he saw Creed I, II, and III and is living it now. He's a contender, to quote Marlon Brando.
When the WBC President made those comments, Paul was a cruiserweight. So he put on another 30 lbs of muscle and fought Mike Tyson. Wow.
I went to my local Gym today, inspired. And I think the people overanalyzing this fight really need to go train more and talk less. Not you, I understand where you are coming from, you clearly have some experience.