It is assumed that Jesus was a Jew. I am not asking whether Jesus was a Jew or not (as we all know that Jesus was born in a "Jewish" household), but does Jesus explicitly call himself a Jew? Rather than his audience thinking he is a Jew, what exact word(s) did he use to describe the religion he followed? Moreover, as there were many Jewish sects at that time did he show affinity to some Jewish sects over the others?
In response to the part of your question about what word Jesus used to describe the religion he followed, I would have to say that it cannot be described in a single word what religion Jesus was aligned to, although, for the sake of compartmentalising Jesus as part of one sect or another, I think @JudahHimango does label Him sufficiently.
To answer the question about what religion Jesus associated himself with; According to John 5:19, Jesus was following what He saw the Father do;
Jesus's religion was therefore not one of the established theologies of the time, but rather a lifestyle of submitting to the will of the Father. Jesus had such a close and intimate relationship with the Father that Jesus understood the character of the Father well enough that He could operate as the Father would do in the same situations.
In John 4:21-26 Jesus identifies himself as a Jew:
Jesus states, that salvation comes from the Jews, the Samaritan woman correctly responds and explains that she knows the Messiah is the cause of that salvation and Jesus goes on to claim to her that He is in fact the Messiah and is therefore claiming to be a Jew.
I am sorry, but I cannot clarify an affinity toward a specific sect, because, maybe my opinion is clouded too much by my own views that Jesus's mission was to show the way not follow a specific way. Jesus had expressed many views which contradicted the main sects such as the time when He set the Sadducees against the Pharisees at the mere mentioning of resurrection.
I cannot find evidence to support my belief, but I am of the opinion that when Jesus talks of salvation coming from the Jews, He is referring to one of the following two descriptions of a Jew, of which I lean toward the latter:
1) A cultural Jew, someone who is of Abrahamic descent.
2) A Jew in the sense of someone who lived according to the will of God, as in the expression that Abraham demonstrated, who had a relationship and trust in God (just like Jesus in my first paragraph), to the extent he was willing to obey God even to sacrifice his own son Isaac. This kind of living is what I would consider the form of 'Jewishness' Jesus understood himself to follow (I know 'Jew' to be a term coined well after Abraham, with a completely different meaning).
I do not think Jesus was referring to a specific sect of Jew in terms of which theology they followed.
Was Jesus a Jew?
Yes. He was born and raised up in a Jewish family and that was also from the descendant of King David.
Did Jesus follow Judaism?
Yes. He was circumcised. He followed the Torah perfectly. In fact, He knew the Torah better than anyone else. He was called Rabbi, a title given to those who teach from the Torah.
Jesus was more than a Rabbi
Jesus came to this world to fulfill the prophesies of the Torah that the Messiah would come. Jesus Christ is the Messiah promised by the Torah and He came to complete the Tabernacle System demanded by the Torah by making himself a sacrificial lamb for the sins of mankind.
Jesus Christ is the Word of God, the promised Messiah who was with God from the beginning even before the creation of the world, and was born as a man without sin to die for our sins, in order to become the perfect Lamb for the sacrifice.
What is the exact word he used to describe the religion he followed?
Jesus did not come to follow any religion but to make a new covenant between man and God, thereby giving a whole new meaning to the true religion or true worship.
Jesus came to make a completion to the Law of Moses, which was in fact given by Himself. God set a standard that man should follow. Since man cannot complete the Law, God himself came as Jesus Christ to complete the Law He himself had made. The Law of Moses demands the Blood of an innocent lamb for the forgiveness of sins. Jesus became the innocent Lamb and sacrificed himself to God in fulfillment of the Law. Since Jesus had to complete the Law, he had to follow the Law, which in turn made him appears to follow Judaism religion. In contrast, He is the author of Judaism religion and the creator of the new covenant, Christianity.
Yes, Jesus was a Jew who practiced Judaism, the religion of the Bible in the 1st century. The gospels record Jesus teaching the Torah and prophets at the Temple, was called 'rabbi', kept disciples (common in 1st-century Judaism), discussed matters of the Torah (the Law) and made halachic rulings on it (e.g. Matthew 5:17-30)
Did Messiah follow a particular sect of Judaism? Yes and no.
Jesus did have differences with the 1st-century religious leaders, of course. Of the 3 major sects of 1st-century Judaism -- Pharisees, Sadducees, Essenes -- Messiah's teachings most closely resemble the Hillel submovement of Pharisees, who believed in the resurrection, a more accessible approach to Torah, and a philosophy from its founder that closely aligns with what Messiah said were the 2 greatest commandments:
This very much aligns with Messiah's statement about the greatest commandments in the Torah:
Hillel Pharisees took an approach to Law-keeping that favored pragmatism over religious stringencies, and this group was generally popular with the common people.
Contrast this with the harshness of the Shammai Pharisees, who suggested zero commerce and communication with gentiles. Or Sadducees, an elite scribal sect who rejected an afterlife, resurrection, and rewards and penalties after death. Or the violence of the Zealots whose murderous acts in the Temple courtyards are thought to be a reason for the Temple's destruction in 70CE.
Messiah didn't align with Hillel Pharisees in all matters. For example, Messiah's judgment on the divorce -- that it should only be allowed in extreme cases -- was the position of the Shammai Pharisees, whereas Hillel Pharisees thought a man could divorce for a variety of reasons. And Messiah's teachings on abstinence from worldly pleasures, voluntary poverty (e.g. "sell all you have and follow me"), and ethical uprightness more closely resembles the Essene sect than Pharisee groups.
Bottom line: it would not be proper to say that Messiah belonged to a particular sect of Judaism. While he most closely aligned with Hillel Pharisees, we believe he's the Messiah; if he happened to align with one sect of Judaism more than another, that says something about the righteousness of that sect more than Messiah's perceived religious alliances.
Let me append to what Mawia said.
Jesus clearly identified with the Jews of His day, although correcting its errors in the process. God purposely sent Him to Judah:
And He clearly said:
Jesus was an observant of Judaism. Both of Jesus' parents had “done everything required by the Law of the Lord” (Luke 2:39). His aunt and uncle, Zechariah and Elizabeth, were also Torah-observant Jews (Luke 1:6), so we can see that probably the whole family took their Jewish faith very seriously.
In the Sermon on the Mount (Matthew 5–7), Jesus continually affirmed the authority of the Torah and the Prophets (Matthew 5:17) even in the Kingdom of Heaven (Matthew 5:19-20). He regularly attended synagogue (Luke 4:16), and His teaching was respected by the other Jews of His day (Luke 4:15).
Even though Jesus was a Jew, Christians are not required to follow Judaism. The Laws of Judaism were given to Moses for the children of Israel in a very sacred and special covenant at Mount Sinai. But this wonderful covenant was only a picture of a New and better covenant that God would one day give to His people, both Jew and Gentile.
This new covenant is recorded for us in Jeremiah 31:31-34:
Thus Mosaic covenant has been fulfilled in Jesus Christ and that old covenant has been replaced. Jesus Himself said:
Jesus was born in Judah, to parents who lived the religious life of those who followed the teaching of Torah, which was explained to all and sundry through the teaching of some 20,000 translators who were scribes and/or Pharisees. We know he was circumcised on the 8th day. We know Jesus supported the teachings of the Pharisees, and told his disciples to do whatever they taught from "Moses seat" in the synagogue.
“Then spake Jesus to the multitude, and to his disciples, Saying, The scribes and the Pharisees sit in Moses’ seat: All therefore whatsoever they bid you observe, that observe and do ...” Matthew 23:1-3a, KJV.
He attended a synagogue daily, attended all religious festivals and observed Shabbat. Nothing He said ever contradicted the teachings given to Moses on Sinai for "Israel."
“For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled. Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven. For I say unto you, That except your righteousness shall exceed the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven.” Matthew 5:18-20, KJV.
So, from his behavior, we know He was an observer of Torah - not withstanding the attempt by the Greek writers of the NT to show Him breaking "the Law." He was not a "jew," - this word did not enter the lexicon until the middle ages and was a pejorative (swear word.) Did He prefer one sect over another? He told his disciples to keep out of Samaria, and to not "go in the way of the gentiles," (which had the meaning then of any nation that was not descended from Abraham.)
“These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not: But go rather to the lost sheep of the house of Israel.” Matthew 10:5, 6, KJV.
His primary dealings were with the followers of Torah - "Israel." He came for the "lost sheep'" who were those Torah followers who had lost their trust in God. The Samaritans and the other sects were not Torah followers, though they also believed in Jahweh. But as Luke said "He died for the sins of the whole World." All benefit from this event, and He did not withhold his love from anyone - Christian or not.
Yes Jesus was a Jew. Not only was he born into a Jewish family and Jewish by birth right, we see that he is not what you might now call a "secular jew" in the modern day sense. No, he respected the law and it's purpose as we see here:
Which means he placed himself under the law and so obeyed it's demands (note: not necessarily Jewish traditions).
While there might have been "Jewish religions" or secs back then. Jesus followed only the one true religion which was what God told him to do.
Along with this we see that he was called a Rabbi and had followers and taught in the synagogues. Absolutely - He was a Jew!
I hope that makes it clear to you.