Israel is the Birthright name, i.e., God's Birthright to His chosen elect.
Each family who has a firstborn son, the father's inheritance belongs to that firstborn son. Yet our Heavenly Father has a Birthright also that goes to His chosen elect, which He counts as a firstborn:
Ex 4:22
22 And thou shalt say unto Pharaoh, "Thus saith the LORD, Israel is My son, even My firstborn:
KJV
That is in the 'to be born of God' sense, called and chosen sense; definitely fit for Jacob having been chosen and having prevailed with The Angel of The LORD for a blessing. But not only did God given Jacob the new name 'Israel', but Jacob also anointed with oil the stone he had made his pillow and dreamed upon, and called it Beth-El, meaning House of God. Wherever Israel went, that stone went with them.
Jer 31:9
9 They shall come with weeping, and with supplications will I lead them: I will cause them to walk by the rivers of waters in a straight way, wherein they shall not stumble: for I am a father to Israel, and Ephraim is My firstborn.
KJV
This above is for in final; notice those "rivers of waters", which are about God's River of the waters of life at the end of Revelation. After God had split old Israel into two kingdoms, Ephraim would represent the northern ten tribe kingdom of "Israel". That is who He is talking about here. This part is not about Judah, not the southern kingdom of Judah. This Jer.31:9 verse is about the ten tribes only, which is why Ephraim is specifically linked here with the name "Israel". Ephraim after the split represents God's firstborn Israel because of God's Birthright promise that wound upon Ephraim per Genesis 48.
However, after God scattered the ten tribe kingdom of Israel, and only Judah remained in the holy land, Judah, or the Jews, began claiming they represent God's Israel. In reality they still are Israelites like all the 12 tribes are. But IF... they continue to reject Jesus of Nazareth as The Christ (Messiah), then they will be 'cut off', and will not be able to claim 'Israel' as their name, simply because the name Israel represents God's Salvation name under His Son, and it involves His Birthright Blessings.
God's Birthright was given to Abraham, and it then went to his son Isaac. At this point there began a contention, as the sons of Ishmael claim that Ishmael, the firstborn son of Abraham and... Hagar (Sarah's Egyptian slave), should be the heir to Abraham. But God chose His Birthright to the Seed of Abraham and Sarah, which was Isaac.
Then the Birthright went to Isaac's son Jacob. A second contention happened with Jacob because his brother Esau was actually Isaac's firstborn. But God had chosen Jacob before the children were born.
Then the Birthright went to Jacob's son Joseph, which began yet another contention. Jacob's son Reuben was the firstborn, but he defiled his father's bed by sleeping with one of Jacob's concubines. So the Birthright went to Joseph.
This began yet another contention between Joseph and his eleven brethren who were jealous. They kidnapped Joseph, put goat's blood on his coat, and acted to their father that Joseph had been killed. Midianite merchants seized Joseph and sold him into Egypt into slavery. God exalted him in Egypt and Joseph had two sons born him in Egypt, Manasseh the elder and Ephraim the younger.
Then the Birthright went to Joseph's younger son Ephraim, where it stopped. God's Birthright Blessing is no more transferred in His Word after this point. One of the prophecies involving Ephraim given in Genesis 48 is that Ephraim's seed would become "a multitude of nations":
Gen 48:16-20
16 The Angel which redeemed me from all evil, bless the lads; and let my name be named on them, and the name of my fathers Abraham and Isaac; and let them grow into a multitude in the midst of the earth.
17 And when Joseph saw that his father laid his right hand upon the head of Ephraim, it displeased him: and he held up his father's hand, to remove it from Ephraim's head unto Manasseh's head.
18 And Joseph said unto his father, Not so, my father: for this is the firstborn; put thy right hand upon his head.
19 And his father refused, and said, I know it, my son, I know it: he also shall become a people, and he also shall be great: but truly his younger brother shall be greater than he, and his seed shall become a multitude of nations.
20 And he blessed them that day, saying, In thee shall Israel bless, saying, God make thee as Ephraim and as Manasseh: and he set Ephraim before Manasseh.
KJV
Joseph tried to correct placement of his father Jacob's hands upon his two sons Ephraim and Manasseh. The right hand is the Birthright blessing hand, so the elder firstborn son is presented on the right-hand side. But Jacob per God's direction crossed his arms, and put his right hand on the younger son Ephraim's head who was on Jacob's left. So Jacob gave Ephraim the Birthright transfer.
1 Chron 5:1-2
5:1 Now the sons of Reuben the firstborn of Israel, (for he was the firstborn; but, forasmuch as he defiled his father's bed, his birthright was given unto the sons of Joseph the son of Israel: and the genealogy is not to be reckoned after the birthright.
2 For Judah prevailed above his brethren, and of him came the chief ruler; but the birthright was Joseph's:)
KJV
Just as Joseph was separated from his brethren, Ephraim and the majority of the tribes of Israel with him was separated from the house of Judah at Jerusalem. To this day, the majority of Ephraim and the ten tribes have yet to return to the holy land. That will not occur until Christ's day of return. Ezekiel 37 is a revelation of this gathering when Jesus returns, including the re-establishing of David's throne in Jerusalem.