What relationship does the nation of Israel have to Jewish identity and faith?
The nation of Israel is a Jewish state. Historically, the name Israel comes from the twelve tribes of Israel. Each of the tribes was headed by and named for a son of Jacob (who was renamed Israel after an encounter with an angel). During this time, the people were called Hebrews.
Jacob moved his family from Canaan, roughly the area of modern-day Israel, to Egypt to avoid famine. Many years later, Moses, a descendant of Jacob's son Levi, led the Hebrews out of Egypt into the promised land. They named this land Israel, and the people were called Israelites.
The term "Jew" came into use around 900 BCE (see the links below). When King Solomon died, conflict arose among the tribes, and the kingdom split in two to form the northern kingdom of Israel and the southern kingdom of Judah. The people who lived in Judah, later Judea, were called Judahites or Judeans. Thus they began to be called Jews and their religion was Judaism.
The modern nation of Israel is composed of Jewish people whose families have always lived in the Middle East and Jews whose ancestors had been forcibly removed to other nations or who had settled in other countries. Many residents of Israel are Holocaust survivors or their children.
Few modern Jews know what tribe they belong to. Today they identify themselves by religion. They practice Judaism; therefore, they are Jews.
Read the books of Genesis and 1-2 Chronicles.