Who is the true Jew? Those who live by faith are the children of Abraham - The Children of the Promise.
What is the essence of being a true Israelite? It is not literal descent from Abraham, but faith in God. It's in the Bible: "Know ye therefore that they which are of faith, the same are the children of Abraham." Galatians 3:7. "For they are not all Israel which are of Israel; neither because they are the seed of Abraham are they all children… but the children of promise are counted for the seed." Romans 9:6-8.
The children of promise are those who put their faith in Jesus. It's in the Bible: "And if you are Christ's, then you are Abraham's seed and heirs according to the promise" Galatians 3:29.
Whether they are Jewish or Gentile, God regards only those who have renewed hearts as true Israelites. It's in the Bible. "For he is not a Jew which is one outwardly, nor is that circumcision which is outward in the flesh; but he is a Jew which is one inwardly, and circumcision is that of the heart, in the Spirit, and not in the letter' whose praise is not from men but from God." Romans 2:28,29, NKJV. "For we are the circumcision, who worship God in the Spirit, rejoice in Christ Jesus, and have no confidence in the flesh." Philippians 3:3, NKJV
God has a special covenant with those who are true, spiritual Israelites. It's in the Bible: "For this is the covenant that I will make with the house of Israel… says the Lord, 'I will put My laws in their mind and write them on their hearts; and I will be their God, and they shall be My people. None of them shall teach his neighbor and none his brother saying, "Know the Lord," for all shall know Me, from the least of them to the greatest of them. For I will be merciful to their unrighteousness, and their sins and their lawless deeds, I will remember no more." Hebrews 8:10-12. Compare with Jeremiah 31:31-34 and Ezekiel 36:26-28.
The Bible tells us the spiritual. characteristics of God's spiritual Israel. They are a morally clean people. It's in the Bible: "Truly God is good to Israel, to such as a pure in heart." Psalm 73:1. They are a people who are spiritually intimate with God. It's in the Bible. "He has exalted the horn of His people, the praise of all saints, even of the children of Israel, a people near to Him." Psalm 148:14.
They are a completely honest people. It's in the Bible. "I will mention the lovingkindness of the Lord … toward the house of Israel, which He has bestowed on them according to His mercies, and according to the multitude of His lovingkindnesses. For He has said, Surely they are My people, children that will not lie: so He was their Saviour." Isaiah 63:7,8. Speaking of Nathaniel, Jesus declared, "Behold an Israelite indeed, in whom is no guile!" Jn. 1:47. Also see Zephaniah 3:13.