ANSWER: No, he did not have a last name like us in the 21st century. Although many people refer to Him as Jesus Christ, the Greek word translated as "Christ" is Christos (Strong's Concordance # G5547), which means "anointed." It was commonly used to refer to the prophesied Messiah or Savior. It would be more proper, grammatically, to say the phrase "Jesus, the Christ."
Note what the archangel Gabriel stated to Mary regarding his name when he annouced that she, among all women, would be the one to give birth to God's son.
30. Then the angel said to her, "Do not be afraid, Mary, because you have found grace with God; 31. And behold, you shall conceive in your womb and give birth to a son; and you shall call His name Jesus. 32. He shall be great, and shall be called the Son of the Highest; and the Lord God shall give Him the throne of David, His forefather (Luke 1:30 - 32, HBFV)
The word Jesus is a transliteration of the Greek word which is listed as Strong's #G2424. A transliteration is taking the way a word is pronounced in one language and spelling it according to the way it SOUNDS when written with the letters of another language. The Greek word, in turn, is derived from the Hebrew or Aramaic word Yehowshuwa. This word, Strong's #H3091, is the same Hebrew word translated in the Old Testament as Joshua.
It is interesting that the name of the God of the Old Testament, first mentioned in Exodus 6:2, is the Hebrew word YHWH (Jehovah, Strong's #H3068). The YHWH (or YHVH) is commonly referred to as the Tetragrammaton.
And God spoke to Moses and said to him: 'I am the Lord (YHWH). I appeared to Abraham, to Isaac, and to Jacob, as God Almighty (El Shaddai, Strong's #H410 and #H7706), but by My name Lord (YHWH) I was not known to them . . . ' " (Exodus 6:2 - 3)
Note that Hebrew does not print the vowels in words as is done in English. If you write God's name as it was written in the Old Testament (we would pronounce it Yahweh or Jehovah) and add the Hebrew word we would pronounce yasha (which means "salvation, or "to free"), you will get the word Yehowshua.
This clue to the identity the Lord as the SAME GOD who created heaven and earth, who wrote the Ten Commandments with His finger, and who dealt with the nation of Israel is sadly hidden from many people.
Although Jesus did not have a last name like us, the Bible DOES refer to him by various titles, designations and descriptions including Emmanuel (Isaiah 7:14, Matthew 1:23), Good Master (Matthew 19:16), Lord of Lords (Revelation 17:14; 19:16), Prince of Peace (Isaiah 9:6), the Messiah (John 1:41), the Holy one of Israel (Isaiah 41:14; 54:5) and many others.