Is the Israel in the Bible the same Israel we know today?

In terms of genealogy, they are the same Israel as in the Old Testament. In the Bible, they are the generations of the twelve tribes of Israel. However, Israel's belief system started to evolve during the time of the New Testament in the person of the religious authorities: scribes, Pharisees, Herodians, chief priests, elders, and Sadducees. In contrast to Moses, Isaiah, David, and the other prophets of the Old Testament, they expected a human political figure to deliver them from their colonizer. They anticipated a human rescuer more than a divine one. This is the current theological perspective of modern Jews. The Father is the only deity recognized in modern Israel, which is more Unitarian. This is not the same faith as the Israelites of the Old Testament prior to Jesus, who acknowledge both the Angel of the LORD and the Spirit of God as Yahweh. 

Nevertheless, in defiance of Israel's unfaithfulness, the Lord continues to protect and preserve them as a chosen nation in the midst of their surrounding enemies. They will one day see the One they pierced and crucified. In order for them to have the opportunity to confess Jesus Christ as Lord (Yahweh), to the glory of God the Father, it is crucial that we proclaim the gospel and disciple the nation of Israel. 

Behold, he is coming with the clouds, and every eye will see him, even those who pierced him, and all tribes of the earth will wail on account of him. Even so. Amen. I am the Alpha and the Omega,” says the Lord God, “who is and who was and who is to come, the Almighty.” Revelation 1:7-8

Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age.” - Matthew 28:19-20