Christ (Christos) is not a surname nor a personal name, but a Greek translation of the Hebrew title Mashiach (Messiah in English), meaning “the Anointed One.” Therefore, the name Jesus Christ is a shortened way of identifying Jesus of Nazareth as “Jesus, the Christ.”
Although Last Names are a rather modern innovation, in ancient Hebrew society it was common for people to use a patronymic, meaning that they were identified as the child of their father. So, for example, Jesus was commonly known during his earthly life as “Jesus, son of Joseph” (cf. Matt. 13:55).