No, one does not need to trace his ancestry back to Biblical times in order to be Jewish. Many of today's Jews, whether they know it or not, are descended from converts to Judaism somewhere along the way.
And one does not necessarily have to be a "follower of the Jewish faith" to be Jewish, since Jews are both a national/ethnic group and a religious group, as distinguished from, e.g., Christians and Muslims. However, if one born to a Jewish mother formally converts into another religious tradition, he would no longer be considered Jewish according to Jewish law.
“Jewish mother formally converts into another religious tradition, he would no longer be considered Jewish according to Jewish law.”
Not according to Orthodox. He’s still Jewish, just heretical.