There were both schools for children and universities for higher learning in the first century CE. Jesus lived in a Jewish nation that was controlled by the Roman Empire. Jewish people, depending on wealth and status, would have had access to traditional religious education and the Roman form of schooling, which was based on Greek education.
In Jewish education, school began for boys at the age of 5. Early education would have consisted of learning to read the Torah and learning to write by copying verses. At the age of 10, boys began studying Jewish law under the instruction of a rabbi, who taught them Jewish law, or Torah, the first five books of the Old Testament. Formal education ended for most by the age of 18, when they would apprentice for a trade or continue on to university to learn to become scribes. Girls were trained at home by their mothers or other women. All Jewish education had a religious base.
Roman/Greek education focused on science, philosophy, and the arts. Children were usually trained by tutors in their homes; archaeologists have found what they believe are tablets that children used to practice their writing. Later they could attend university for specialized training in law, science, medicine, and the like. One of the most famous medical schools in the world at the time was in Alexandria, Egypt.
There was definitely schooling in Jesus' time, though it took different forms than now, and had had a different focus. Training in daily crafts would be passed down from father to son/mother to daughter, and so would not be formal. The closest thing to formal education would be religious education, which would take place at the temples, through spoken discussions with masters of the scripture. As far as the term "Jesus the Jew," he was born and raised Jewish, and was educated in Jewish religious tradition. Legally, he was Jewish.