This is quite a question! In fact it is such a question that the only answer lies in whoever happens to be giving the explanation.
As offensive as it may be, there are so many Gentiles wanting so bad to be Jewish that they have come up with all kind of replacement doctrines, inclusion doctrines, etc..
How many of these people, watching their kids and grandkids being grabbed by Nazis, would have not said, "Okay, so we are not R E A L L Y ... S O sure that we are Jewish?"
By the explanation of the word "Jew", being derived from the word "Judah", Abraham, Isaac and Jacob were not Jewish, but David, Solomon, Jehoshaphat, Caleb, and Yeshua are Jewish.
Most people today refer to a person born of a Jewish mother and father as Jewish. Various exceptions are;
Israel is now allowing people whose father is Jewish to gain citizenship, although in most circles, it was a person's mother's religion that has been the determining factor in whether a person was Jewish or not.
Many Jewish people, even those who attend synagogue less than a dozen times a year, do not consider Messianic Jews to be Jewish.
In Hitler's math, one fourth Jewish blood was enough for the gas chambers and other horrors.
Conversion is another interesting subject.
Orthodox Jews have attempted to make the Law of Return void in the cases of Gentiles converted by non-Orthodox methods.
A convert to Judaism, regardless of what comes out of the mouths of "real" Jews, will never be accepted as a "real" member of the tribe.
One major concept which can't be denied, is that anyone celebrating the Jewish holidays instead of the Gentile ones, keeping biblical kosher, and other commandments are Jewish by religion and lifestyle. They have chosen to live a Jewish lifestyle and to bring their kids up in a Jewish home environment. A second or third generation Messianic Gentile, could honestly say that their home and upbringing was Jewish, and that they were raised with Jewish customs and traditions, but they do not know if they Jewish blood.
Unless there is absolute proof that a characteristic of DNA can absolutely prove someone's Jewishness and a DNA test given proves the person to be Jewish, we, who have "Jewish" parents, may just be as Gentile as our Gentile neighbor!