Being that a state has rules for using the name "University" and Trump called his scam such a name, one has every right to think it was an University. Since by law the name meant something. If it didn't mean something why did Trump change it after switching states where the home office was to try and buy more time.
As far as nobody claiming it, just by default the name implies it. Trump University wasn't chosen to give the appearance of a scam for obvious reasons.
The Company was an LLC. That’s the name they chose. There is no such rule for the use of the word University. He changed it because he basically got blackmailed into it by the judge.