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.