This is very mistaken.
I'll give you an analogy. John Doe lives in America. He is an American. He also lives in Texas. He is a Texan. He also lives in Behar County. He is a resident of Behar County. He lives in Houston. He is a Houstonian.
Abraham the Hebrew was the father of the Children of Israel. His grandson, Jacob, was renamed Israel by God. So the descendants of Israel were called Israelites and they lived in the land of Israel.
Jacob had 12 sons who became the heads of the 12 tribes of Israel. One of these tribes was Judah which was allocated the land in the region surrounding Jerusalem.
After the reign of Solomon, the kingdom of Israel split into 2 kingdoms. The southern kingdom was known as Judah and incorporated the tribe of Judah and Benjamin, as well as some miscellaneous Israelites from other tribes.
The northern tribe was called Israel or Samaria after its capital, Samaria.
The Assyrian armies came against Israel and carried them away into captivity in Israel. They resettled them in their lands and brought in captives from other countries and settled them in the region of Samaria. These people intermarried with the Israelite people who remained and produced the half-Jewish race known as the Samaritans, about whom Jesus told the parable of the Good Samaritan.
The southern kingdom of Judah remained until 586 B.C. when it was taken into captivity in Babylon. 70 years later they were allowed to return to the land and rebuilt the temple.
They then were under the rule of the Greek Selucids and then the Romans. During that period the region became known as Judea, after Judah.
During the Medo-Persian reign which followed the Babylonian reign, the Hebrews/Israelites began to be called Jews, after the tribe of Judah. You can see this in the book of Esther.
Now to bring it up to the time of Jesus.
The lineage in both Matthew and Luke trace his line directly from Abraham, clearly establishing that he was descended from Abraham the Hebrew. Further he was descended from King David of the tribe of Judah, which is why Jesus is called the Lion of the Tribe of Judah.
Jesus grew up in the area around the Sea of Galilee. Therefore he was called a Galilean.
He also was from the town of Nazareth and thus was called a Nazarene.
So He was an Israelite/Hebrew descendant of Abraham, through the tribe of Judah, known in his contemporary time as a Jew, from the Galilean region and the town of Nazareth.
Edom was the nickname of Esau, Jacob's brother. The Talmudic rabbis identified Edom as Rome although it was more of an allegory as there is not any Edomite blood to speak of in Romans. It was just based upon the enmity between Esau and Jacob.
The Edomites were never considered Jews. Only Herod and his family who converted so they could be rulers under the Greek Selucids.