ANSWER: Yes, Jesus was a Jew. The word "Jew" is a shortened form of the word "Judean" referring to the descendants of the patriarch Judah. Judah was one of the twelve sons of Jacob (Genesis 35:22-26). Jacob's name was changed, by God, to Israel (Genesis 32:28). Therefore Judah and his descendants formed one of the twelve tribes of Israel. In the New Testament genealogies (Matthew 1 and Luke 3), both Mary and Joseph are listed as descendants of Judah, through the line of David. Not only was Jesus a descendant of Judah, but He was a descendant of the Davidic (or kingly) line of Judah.
The Scriptures contain many references to Jesus being the son (descendant) of David. It was prophesied that Shiloh would come from the tribe of Judah: "The scepter shall not depart from Judah, nor a lawgiver from between his feet, until Shiloh [the Messiah; or Prince of Peace] comes; and to Him shall be the obedience of the people" (Genesis 49:10). Jesus referred to Himself as a Jew to the woman at the well: "You [Samaritans] worship what you do not know; we [Jews] know what we worship for salvation is of the Jews" (John 4:22). He used the pronoun "we" as He clearly considered Himself to be a Jew. However, the Jews of His day rejected Him (John 1:11). The Apostle Paul wrote "For it is evident that our Lord arose from Judah..." (Hebrews 7:14).
In Revelation 5:5 Jesus Christ is referred to as "...the Lion of the tribe of Judah, the Root of David..." God the Father sent "...His own Son in the likeness of sinful flesh, on account of sin: He condemned sin in the flesh" (Romans 8:3). Christ took on the flesh and blood of His physical parent Mary: "Inasmuch than as the children have partaken of flesh and blood, He Himself likewise shared in the same, that through death He might destroy him who had the power of death, that is, the devil" (Hebrews 2:14). Jesus was born as a human being (John 1:14) by a divine begettal (Matthew 1:18-23) into the nation, area, and family of Judah.