Jesus didn't act like a Jew , he didn't know many Jewish customs from what I understand.
He was called a rabbi, but comes across overwhelmingly under-qualified to even be one.
One could argue Jewish by birth on the matriarch side, but that's about it.
The question is , what makes Jesus Jewish least of all to be called King of the Jews?
This post will assume its readers accept the stories of the gospels as fact (even though I think that many of them aren't).
First of all, Jesus was a JEW. He was born of a Jewish mother, making him one of the Children of Israel. That means he would have been bound by the covenant between God and Israel, all 613 commandments.
Second, he was raised as a Jew. His parents took him to the temple for the offering of the first born, and to Jerusalem for his bar mitzvah. It stands to reason that they raised him in Judaism just as they would have any child.
He was called up to read and teach from Isaiah at the synagogue. He would have had to have been literate and learned in Judaism with a good reputation for that to have happened, although obviously he had esoteric opinions.
He was called Rabbi, which (even if it was unofficial) indicates he had a very high degree of learning in both written Law and Oral Law. After all, the primary job of a Rabbi is to teach and rule on matters of Law.
He observed Jewish Holy Days such as Chanukah (Feast of the Dedication) and Passover.
He kept the Shabbat, although he participated in the arguments of his day about how best to keep it.
He washed his hands before meals, showing that he observed a certain rendition of the Oral Torah. (It was only "some of his disciples" which did not. Furthermore, this tradition was not yet set.)
In the sermon on the mount, he replied to the rich man that to gain eternal life, one must keep the commandments.
In Matthew 23:1-3 He tells his followers to do and observe everything the Pharisees teach (that would include Oral Torah as well as Written Torah).
In Matthew 23:23 He tells the Pharisees to observe the basics of the Written Law first, and then to observe Oral Law as well.
That's just tooooooo many things lined up for him being an ordinary Jew. His teachings line up with the school of Hillel, which Jews reading the gospels can see. (See Harvey Falk, "Jesus the Pharisee."