Did Jesus himself assert the fact that he belonged to some religion like Judaism?

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:

"Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil" - Matt 5:17.

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.

"I can of my own self do nothing: as I hear, I judge: and my judgment is just; because I seek not my own will, but the will of the Father who has sent me." - John 5:30

and

"For I have not spoken of myself; but the Father who sent me, he gave me a commandment, what I should say, and what I should speak." John 12:49

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.

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.

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:

John 1:11-12 He came to what was his own, but his own people did not receive him. But to all who have received him – those who believe in his name – he has given the right to become God’s children

And He clearly said:

John 4:22: You people worship what you do not know. We worship what we know, because salvation is from the Jews.

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:

“‘The time is coming,’ declares the LORD, ‘when I will make a new covenant with the house of Israel and with the house of Judah. It will not be like the covenant I made with their forefathers when I took them by the hand to lead them out of Egypt, because they broke my covenant, though I was a husband to them,’ declares the LORD. ‘This is the covenant I will make with the house of Israel after that time,’ declares the LORD. ‘I will put my law in their minds and write it on their hearts. I will be their God, and they will be my people. No longer will a man teach his neighbour, or a man his brother, saying, “Know the LORD,” because they will all know me, from the least of them to the greatest,’ declares the LORD. ‘For I will forgive their wickedness and will remember their sins no more.’”

Thus Mosaic covenant has been fulfilled in Jesus Christ and that old covenant has been replaced. Jesus Himself said:

Matt. 5:17“Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them”.

Yes, Jesus was a Jew who practiced Judaism, the religion of the Bible in the 1st century. The gospels record Jesus teaching the Torah (Law) and prophets at synagogues on shabbat (e.g. Luke 4:16) and at the Temple complex in Jerusalem. He was called 'rabbi', kept disciples (common in 1st-century Judaism), discussed matters of the Torah and made halachic rulings on it (e.g. Matthew 5:17-30).

Did Jesus follow a particular sect of Judaism? Yes and no.

Jesus did have differences with the 1st-century religious leaders, of course. Of the major sects of 1st-century Judaism -- Pharisees, Sadducees, Essenes, Zealots -- Jesus' 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:

"That which is hateful to you, do not unto another: This is the whole Torah. The rest is commentary — [and now] go study."

-Hillel the Elder, founder of the Hillel Pharisees

This very much aligns with Jesus' statement about the greatest commandments in the Torah:

You are to love the Lord your God with all your heart and with all your soul and with all your strength.’ [Deut 6:5] This is the greatest and most important commandment. And a second is similar to it, ‘You are to love your neighbor as yourself.’ [Lev 19:18] All of the Torah and the Prophets are dependent on these two commandments.”

-Jesus

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.

Jesus didn't align with Hillel Pharisees in all matters. For example, Jesus' 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 Jesus' 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 Jesus 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.

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.

John 3:1-2 (NIV) Now there was a Pharisee, a man named Nicodemus who was a member of the Jewish ruling council. He came to Jesus at night and said, “Rabbi, we know that you are a teacher who has come from God. For no one could perform the signs you are doing if God were not with him.”

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.

Matthew 22:41-46 (NIV) While the Pharisees were gathered together, Jesus asked them, “What do you think about the Messiah? Whose son is he?” “The son of David,” they replied. He said to them, “How is it then that David, speaking by the Spirit, calls him ‘Lord’? For he says,

“‘The Lord said to my Lord: “Sit at my right hand until I put your enemies under your feet.”’ (Psalm 110)

If then David calls him ‘Lord,’ how can he be his son?” No one could say a word in reply, and from that day on no one dared to ask him any more questions.

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.

John 1:14 (NIV) The Word became flesh and made his dwelling among us. We have seen his glory, the glory of the one and only Son, who came from the Father, full of grace and truth.

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.

Hebrews 12:2 (NIV) Fixing our eyes on Jesus, the pioneer and perfecter of faith. For the joy set before him he endured the cross, scorning its shame, and sat down at the right hand of the throne of God.