Can't remember if the egg is static and the 'stamp' moving horizontally, or the stamp is static and the egg rolls over it. Anyway, the end result is the same, effectively the egg rolls over the stamp and 'picks up' the ink image.
Its sprayed on!! Works an ink jet printer (with the paper moving instead on the nozzle) where the egg moves very close past an ink nozzle at a set speed and the ink is shot out the nozzle in a set pattern that results in the lion mark (or what ever).