what would happen if the coast guard would board your boat and found you had no
proof of ownership or documentation?
Pleasure boats must be either state registered or Coast Guard (Federally) documented ; according to tonnage.
Boats with state registration are required to display state registration numbers, and carry state registration document.
Failure to comply with either requirement is a misdemeanor and operator is subject to being fined much like a motor vehicle.
Documented vessels are not required to display exterior marking numbers, but are required to display boat name and "home port", usually on the stern, and to carry their current CG documentation papers.
Having been stopped and boarded twice (in the Ches. Bay) for "safety inspections", and, having been intercepted (but not actually boarded) twice (at night off SE Florida Coast) by the Coast Guard, I can tell you that as a practical matter, once given the boat name, home port, and owner's name, etc., ... they have ALL of your information in their on board computer databases and don't actually have to see the piece of paper.