The word Christ is not a personal name but a title in Greek that means "the Anointed One." In Hebrew, the title is Mashiach, or, more commonly, Messiah. The name "Jesus Christ" is a shorthand way of identifying Jesus of Nazareth as "Jesus, the Christ."
In ancient Hebrew society, people used patronyms, meaning that they were identified as the child of their father. So, for example, Jesus was known during his earthly life as "Jesus, son of Joseph" (cf. Matt. 13:55). St. Joseph and the Blessed Virgin in turn would have been identified as the son and daughter of their respective fathers.