The United Nations keeps a registry and a data center at NASA assigns designations, but only communications satellites are subject to practical controls.
Every satellite must be registered to a particular country, for reasons of liability under space law.
A registry has been maintained by the UN Secretariat since 1962, and another registry, established under a 1976 agreement, is kept by the agency’s Committee on the Peaceful Uses of Outer Space.
Registry listings include a designation, determined by the World Data Center for Rockets and Satellites at NASA/Goddard Space Center, for each satellite. The listings include details of each object’s orbit, altitude, and function.
But only the International Telecommunication Union coordinates launchings, by assigning the frequency slots for transmissions so that satellites won’t interfere with one another.