As with every other fractional company, when you searched the registration through the FAA database you only get the FIRST name on the registration. The “Fractional Owner” field will say “YES” and the rest of the names are listed at the bottom of the page under "Other Owner Names."
Try it with any NetJets, Flexjet, Flight Options, Citationair, Avantair, or Planesense airplane.
But to really answer your question, the reason it doesn’t come up on FlightAware as an airline because they don’t file via a callsign. They just use the N-Number. There was talk years back about getting an FAA telephony callsign but for some reason it was disapproved. (suggestions included “Granite” (NH=granite state) and my personal favorite “Blue Canoe”)
Rigi isn’t the actual operator. It’s a holding company that manages the unsold shares of new airplanes and the “core” airplanes that make up the rest of the fleet.