Jesus (Yeshu bar Yosef) of Nazareth was not a Jew. He was a Galilean. True, he followed the Torah and the Prophets and the Writings, went to synagogue, attended the temple in Jerusalem for major feast days when possible, and probably wore a tallit with tzizit and tefellin in his wrist and head daily but he was not a Jew. In first century Roman Palestine, what you believed religiously did not matter as much as where you were born (unless you were born in the Diaspora) and to whom you were born.
To be a Jew in Roman Palestine meant first of all to be from Judea, the core Judea, not the expanded version which took in Samaria, Philistia, Idumea, and Nabatea. Jews, or Judeans, looked down their nose at Galileans almost as much as they did at Philistians, Idumeans, and Nabateans, though not as much as they looked down on Samaritans. The name Galilee derives from Galil ha-Goyim, “District of the Gentiles”, and its population was a mix of forced colonists deported by the Hasmoneans, descendants of converts, and Gentiles.
Before its conquest by the Hasmoneans, Galilee had been the southern territory of the Arabic kingdom of Iturea in much of southern Syria, whose territory was conquered by Aristobolus I along with Galilee, with the requisite, for Hasmonean tyrants, forcible conversion of the populace. There is not inconsiderable chance that Jesus could have been part Arab.
In addition, one had to be not just from Judea but of the correct ethnicity, because in the first century Judea took in the formerly independent territories of the Idumeans, Nabateans, and Philistians. Those groups followed the Torah, most also the Prophets and the Writings, went to synagogue, and attended major feasts at the temple in Jerusalem, but were not considered entirely kosher because their immediate ancestors had been forcibly converted under the Hasmoneans, who were even more tyrannical and fratricidal than their Herodian successors.
Since we’re on the subject of 1st century ethnic groups in Palestine…
The Samaritans in the middle, between Judea and Galilee, followed the Torah but not the Prophets or the Writings, went to their own synagogues, and attended major feasts in their own temple atop Mt. Gerizim, destroyed in 110 BCE by the Hasmoneans but rebuilt by Herod the Great in 10 BCE.
The main reason for the spiteful, vindictive attitude of most if not all 1st Jews towards the Samaritans had to do with the northern kingdom having been the senior and historically wealthier and more cosmopolitan of the two Hebrew/Israelite kingdoms. Under the Assyrian, Chaldean, and Persian empires, Yehud, the southern realm, was governed as a sub-province appended to and subordinate of Samerina, the northern realm. That situation changed briefly under Alexander, but resumed under the Ptolemies.
In inscriptions and surviving papyri, Samerina is also known as Bit Humria, or House of Omri, while Yehud is also known as Bit Dawid, or House of David, though this junior kingdom was at first simply known as Teman, Hebrew for “the south”.
The Jewish sect of the Pharisees, which also had adherents among the Galileans, used the Torah, the Prophets, and the Writings, and included the Mishna, or Oral Torah, in its teachings. It was divided into two factions itself, Beth Hillel and Beth Shammai, with Beth Hillel the dominant of the two, with its head holding the title of Nasi, head of the Great Sanhedrin.
The Jewish sect called Bene Sedeq accepted the Torah, the Prophets, and the Writings, but not the Mishna; their “descendants” became the Karayim.
The Jewish sect of the Essenes, which probably had adherents in Galilee since it certainly did in Damascus, used not only the Torah, the Prophets, and the Writings, but a number of other sacred scriptures as well. Their main cult center seems to have been the large commune at Qumran, but enough of them visited Jerusalem to have their own gate.
Not all 1st century Jews subscribed to the Prophets and the Writings. The Jewish sect of the Sadducees, like its Samaritan neighbors, accepted only the Torah. However, its adherents worshipped at the temple in Jerusalem and traced their ethnicity matrilinealy like the rest of the Jews as well as the Galileans where the Samaritans traced their ethnicity patrilinealy. Its head was the high priest and necessarily tied to the temple cult.
Hellenistic Jews used the Septuagint, a Greek-language version of the Tanakh translated in Alexandria and used throughout the Diaspora which included several books and passages not found in the Hebrew Tanakh or Aramaic Targum of Palestine. Diaspora Samaritans had their own translation into Greek of the Torah.
The bitter rivalry between Jews and Galileans, even within the same sects, lasted at least through to the end of the Great Jewish Revolt (66-73 CE), or at least the end of its major phase, the fall of Jerusalem. One of the main handicaps suffered by the rebels was the fratricidal infighting among their four or five major factions. Two of these were rival factions of Zealots, one from Judea and the other from Galilee.
The sect of the Sadducees disappeared with the destruction of the temple in Jerusalem in 70 CE, and the Essenes soon after the destruction of their base at Qumran in 68 CE. The Samaritans lost their own temple again in 67 CE, but they did not go away. Of the two power centers of the Jews at the turn of the era, only the Great Sanhedrin in the hands of the Pharisees and Beth Hillel was left after the Jewish Revolt to shape mainstream Judaism for the future.
The distinctions between Jews and Galileans vanished after the Bar Kokbha War 130-135 CE, when nearly all the elite remaining in Judea were deported to Galilee along with the permanent move of the Great Sanhedrin and seat of the Patriarchate to there in 140 CE.
Contrary to popular belief, the Samaritans, who in ancient times were equally numerous with the Jews and Galileans combined, did not simply vanish. Many became Christians or Gnostics, with the latter eventually conforming under the Roman Empire. Following the Muslim conquest in the 7th century, the great majority converted to Islam, especially after the Abbasids gained the Caliphate. Around a thousand Samaritans remain, half in the State of Israel and half in Nablus in the West Bank of Occupied Palestine.