Does every single aircraft have a unique registration mark or number? If, for example, 2000 Boeing 777 units were produced, do they have their own registration number? And how is that registration number regulated? Who keeps the number? What is that registration intended for?
Yes, every aircraft has a registration number--or "tail number" since it is commonly painted on/near the tail. This links each make/model/serial combination to a specific owner, similar to a car's license plate.
Each country assigns tail numbers with that country's unique one-, two- or three-letter prefix(es), e.g. "N" for the USA and "XA" for Mexico, followed by some number of letters and/or digits according to their chosen scheme. Some countries allow requesting a specific number not already in use (or reserved), similar to vanity license plates.
For non-commercial flights, a plane's tail number is used as its radio callsign since it's guaranteed to be unique. Commercial flights typically use the carrier's callsign plus flight number.