Before we can even have an intelligent discussion about Israel, we have to understand there are actually thirteen tribes...
Reuben
Simeon
Levi
Judah
Dan
Naphtali
Gad
Asher
Issachar
Zebulun
Joseph - Who had two sons who were made tribes
Ephraim
Manasseh
Benjamin
How many of these tribes are Jews?
ONE - Judah
Here is an interesting scripture
2Ki 16:1 In the seventeenth year of Pekah the son of Remaliah Ahaz the son of Jotham king of Judah began to reign.
2Ki 16:2 Twenty years old was Ahaz when he began to reign, and reigned sixteen years in Jerusalem, and did not that which was right in the sight of the LORD his God, like David his father.
2Ki 16:3 But he walked in the way of the kings of Israel, yea, and made his son to pass through the fire, according to the abominations of the heathen, whom the LORD cast out from before the children of Israel.
2Ki 16:4 And he sacrificed and burnt incense in the high places, and on the hills, and under every green tree.
2Ki 16:5 Then Rezin king of Syria and Pekah son of Remaliah king of Israel came up to Jerusalem to war: and they besieged Ahaz, but could not overcome him.
2Ki 16:6 At that time Rezin king of Syria recovered Elath to Syria, and drave the Jews from Elath: and the Syrians came to Elath, and dwelt there unto this day.
The context is that Israel is at war with the Jews.
After Solomon death his son Rehoboam became king and wanted to exact terrible taxation on the nation. Jereboam led a breakaway of the the northern ten tribes and established a capital at Samaria. Judah, Benjamin and Levi remained in the south with a capital at Jerusalem. From then on, the southern kingdom was Judah and the northern kingdom was Israel.
All Jews are Israelites but not all Israelites are Jews. Example: All Alaskans are Americans but not all Americans are Alaskans.
Israel went in to captivity to Assyria in 721-718BC. Judah went into captivity to Babylon in 607-587BC.
Also, the name Israel was not to be placed on Judah or the Jews...
Gen 48:3 Then Jacob said to Joseph: "God Almighty appeared to me at Luz in the land of Canaan and blessed me,
Gen 48:4 and said to me, 'Behold, I will make you fruitful and multiply you, and I will make of you a multitude of people, and give this land to your descendants after you as an everlasting possession.'
Gen 48:5 And now your two sons, Ephraim and Manasseh, who were born to you in the land of Egypt before I came to you in Egypt, are mine; as Reuben and Simeon, they shall be mine.
Gen 48:16 The Angel who has redeemed me from all evil, Bless the lads; Let my name be named upon them, And the name of my fathers Abraham and Isaac; And let them grow into a multitude in the midst of the earth."
So Israel (Jacob) said that from then on, the name 'Israel' was to be named on Ephraim and Manasseh. It was not placed on Judah.