This should be a rather straightforward topic but it is funny how often people treat Christ’s full name as ‘Jesus Christ’ when ‘Christ’ isn’t a name, to begin with.
Jesus is the (English) name of the savior. We are given no further information as to whether he had any additional parts to his name. If anything, in that part of the world at the time, it would have been: Jesus, Son of Joseph, or Jesus of Nazareth.
There simply wasn’t the sort of last name pattern so many of us are used to today.
His family was known for what they did. Joseph was a carpenter and thus Jesus was trained in carpentry as well. Or, to be more exact, Joseph was a builder. His profession is often translated as “carpenter” but the term is truly broader and would include working with stone and various materials commonly used in home constructions. As such, Jesus (Yeshua) could also be called, Jesus, Son of the Builder, Joseph.
Christ comes from the Greek (χριστος) and is a title. It is the Greek word for Messiah. In Hebrew, the word would be משיח — (mashiach/mah-shee-ahch – where the “ch” is pronounced hard as in the name Bach – Strongs #4899).*
Many of these examples should make sense because if you stop and think about it, it is also how many of the others referred to in Scripture are named. Some examples:
Paul/Saul of Tarsus or Paul, Apostle of Christ
Ezekiel, the Prophet
Luke, the Physician
Titles were a distinguishing characteristic. This, of course, includes regions of origin as you can see above with Jesus and Paul.
* https://www.ancient-hebrew.org/definition/messiah.htm Retrieved 11.23.2021