Many local police forces (towns, cities, universities, etc.) in the US have this type of register using the serial number but I doubt that it is centrally accessible. Instead, they usually provide a sticker with the police department name/logo that one applies on the frame to identify where it's from. This is certainly not an ideal solution; I would much rather have something like vehicle licensing but free, managed by states or the federal government, which seems more in line with what you mentioned.

Edit: Just found this supposedly national registry: https://www.nationalbikeregistry.com/