Jesus had no last name. Christ is a title that was given to Him. As noted in other questions, "Christ" comes from the Greek word "Christos", which is the translation of the Hebrew word "Meshiach", from which we derive the word "Messiah". So, "Messiah" and "Christ" are transliterated words for the same thing. These words mean "anointed one" in their original languages.
Last names or family names likely had their origin England about 800 years ago, but this was not common in Jewish cultures in the days of Jesus.
However, there were sometimes distinctions made for individuals based on their city of origin or residence. Jesus was sometimes referred to as "Jesus of Nazareth". He was not born there, of course, but that is the place Mary and Joseph lived after their return from Egypt. (Jesus was born in Bethlehem.)
Incidentally, this was the cause of some confusion about whether or not Jesus could be the Messiah, since the people knew that the Messiah would come from Bethlehem and not Nazareth, although another prophecy indicated that He would be called a Nazarene.
As David Morton noted, people were also distinguished by their parentage. This is seen in Simon Peter, in particular, whom Jesus referred to as 'Simon, son of John".
He brought him to Jesus. Jesus looked at him and said, “You are Simon the son of John. You shall be called Cephas” (which means Peter). John 1:42 ESV
In John 1:45, we find both of these distinctions in Philip's reference to Jesus:
Philip found Nathanael and said to him, “We have found him of whom Moses in the Law and also the prophets wrote, Jesus of Nazareth, the son of Joseph.” John 1:45 ESV
Finally, Jesus was also distinguished by his trade. He was known as a rabbi or teacher.
So, there was little need to distinguish between other people with similar names when referring to the "rabbi named Jesus", to "Jesus of Nazareth", or to "Jesus, the son of Joseph".