Ships then, and now, don’t really need numbers for identification. There are various unique numbers that they can and do use occasionally for specific purposes(IMO numbers and hull numbers). However, a ship’s name and home port were, and are, more than sufficient to identify a ship for legal purposes. You don’t need a registration number on a ship, and certainly wouldn’t have needed one then.

The authorities absolutely kept meticulous records of ships entry and exit from any harbour as well as what was on board, what was loaded and unloaded and frequently a list of all persons onboard.

Some flag states enforce uniqueness constraints on name and home port combinations. The US does not, but that really doesn’t matter much in the real world. There just aren’t that many conflicts.

More importantly, the founding fathers very much did not extend privacy rights to ships. Intentionally so. The very first congress passed a law in 1790 that exempted ships from the requirements of needing a warrant to be searched.

The ability to track and search ships without warrants has been an important capability of the federal government from day one.

Hell, the federal register of ships is published and always has been. I don’t know how they would have felt about private cars, but the founding fathers revealed preference is that shipping and ships are not private like your other “papers and effects” are.

Cars, wagons, carts, are not ships