Chapter 28 gives the full account of Jacob being sent to the land of his mother's family to find a wife there; of his older twin-brother Esau despising parental wishes and choosing a wife from Ishmael's clan. But the significance of this with regard to the question lies in chapter 25. There the twins' mother, Rebekah, entreats the Lord, who answers her:
"Two nations are in thy womb, and two manner of people shall be
separated from thy bowels; and the one people shall be stronger than
the other people; and the elder shall serve the younger." Genesis
25:23 K.J.
The account in question cannot be about Jacob becoming 'converted' or, 'a true Jew' any more than it can be about him being converted to faith in Christ, or becoming either a literal or a spiritual spiritual Jew. The Jewish nation no more existed on Earth at the time of Jacob than did Jesus Christ. The account is bound up with Jacob coming to personally know and trust the God of his forebears. It was about Jacob realising the older promise, of how a nation would come from him, blessed of God. That is what is being described in the text in question. He already knew about this God of Abraham and Isaac. The whole family did. That is why, when Jacob set out to obey his father's command to find a wife from his mother's brother's family, his father, Isaac, blessed him with these words:
"And God Almighty bless thee, and make thee fruitful, and multiply
thee, that thou mayest be a multitude of people; and give thee the
blessing of Abraham, to thee and to thy seed with thee; that thou
mayest inherit the land wherein thou art a stranger, which God gave
unto Abraham." Genesis 28:2-5 A.V.
On his way, Jacob had a dream of a ladder set up on the earth, its top in heaven, with angels of God ascending and descending on it. The Lord stood above it, declaring himself to be the God of Abraham and Isaac, promising to give Jacob and his seed the land where he lay. He repeated the God-given promise to Abraham of his seed spreading over the land, and "in thee and thy seed shall all the families of the earth be blessed." God promised this to Jacob, adding that wherever he went, God would bring him back to that promised land. (Genesis 28:11-15)
Jacob awoke, afraid, as he knew the Lord was in that place. He took the stone that had been for his pillows, set it up, anointed it with oil, and called the place Beth-el. Then he vowed to God, as in the verses quoted in the question. All this background is vital if the question is to be answered properly.
The chapters following show what happened over many years when he was tricked into marrying the eldest daughter of Laban, having to then work more years for the younger daughter he loved, Rachel. They both bore him children. When his father-in-law's duplicity turned to anger, God instructed Jacob:
"Return unto the land of thy fathers, and to thy kindred, and I will
be with thee... I am the God of Beth-el, where thou anointedst the
pillar, where thou vowed a vow unto me. Now arise, get thee out from
this land, and return unto the land of thy kindred." Genesis 31:3 & 13
A.V. See also Shared Life, Donald Macleod, pp.12-13 Christian Focus 1994 & Hosea, Michael Eaton, pp.171-176, Christian Focus 1996
The account goes on to show how Jacob returned in faith to the land of his father, along with his wives and children, trusting God to give him that land and to make his descendants into its inheritors. That is what those verses describe. By the time of Moses, his offspring had grown into a very large group known as the Haribu, who did not become the nation of Israel until God, through Moses, released them from bondage in Egypt. Then, at the foot of Mount Sinai, that bunch of despised people became the nation of Israel. That is what all those chapters combine to show as to the significance of Jacob's vow in Genesis 28:20-22.
