A Jew is a Jew by ancestry.
How do you figure to reckon it, though?
The rabbis follow matriarchal bloodlines - i.e. if you're mom's Jewish, you're Jewish. They have done so since roughly the beginning of the 2nd century. But for the entire Old Testament period, bloodlines were patriarchal - determined by the father.
Why did they make that change? It's because patriarchal bloodlines are UNRELIABLE. You can always tell who the baby's mother is. Unfortunately, the same can't be said about fathers. For the Jews, whose history includes centuries of engaging in fertility cults, as well as being deported from their country in antiquity, those patriarchal genealogies are absolutely useless. They are corrupt even from before the times of the earliest ones. And consider the northern kingdom... they were slaughtered wholesale and the remnants dispersed to the four corners of the world. They don't even HAVE genealogies, though they certainly have descendants.
The Jews of the New Testament period knew this. They debated about it. We still have records of the positions taken by Hillel and Shammai with regards to solving the problem. Eventually, the debate was settled and the Jews changed to matriarchal lines.
But that solution does NOTHING to address the problem retroactively. You still have centuries and centuries of unreliable paternity that you're trying to build on. Plus, you have thousands or millions or people who are genuinely Israelite descendants, but don't even know it, having lost their cultural identity.
There was another position suggested by a rabbi during that time period, but Judaism ultimately did not adopt that position. That position was that the true paternity of Jews (and even lost Israelites!) could be determined by observing their behavior. Those who kept the law were Abraham's true descendants. What rabbi who taught such a thing? Jesus of Nazareth.
You see, what you call "spiritual Israel" was not spiritual at all to them in the 1st century. It was a practical, pragmatic approach to solving the question of "who is a Jew?"