How can anyone today, conclusively prove that they are a descendant of Judah, let alone of King David?
As I posted before:
Thus, most Jewish sources agree that the standard is:
- Jews living in Israel, as a whole, desire to return to the Torah as it was given at Mount Sinai. (Torah - written and oral)
- A proven (נביא) prophet in the Jewish community receives something from Hashem that a particular Jewish man is the leader that Hashem has chosen. (Hashem will of course only pick someone who is proven to be an actual paternal descendeant of King David through Solomon.)
- BTW - there are Jews today living in Israel from various communities who have families that reach back to King David through Solomon.)
- A standing Mosaic Court/Sanhedrin is able to clarify the person is fit to be king as specified above.
- Somewhere between the prophet and the Mosaic court an anointing with oil can happen. (The anointing with oil has a specific reason behind it, but it is a long explaination.)
- Said person will receive prophecy from Hashem of what it is they are to do.
- Just below the level that Mosheh ben-Amram (Moses) had.
- Probably on a level above Yehoshua bin-Nun (Joshua).
- Using Torah based wisdom, prophecy, knowledge of science, economics, etc. said Davidic king will be able to lead the Israeli people and bring things back to they way they were between when Mosheh ben-Amram (Moses)/Yehoshua bin-Nun (Joshua) and Shlomo HaMelkh (Solomon) led.
What happened is that Jewish communities always had families who were very particular about record keeping. Thus, in various Jewish communities that have certain names that mark their ancestry.
For example, some Jews with the last name Cohen, Kohen, Kahn, etc. may often carry a family tradition that they descend from Aharon. The rules of who they can marry are very stringent and there are families who were very meticulious about keeping them and many of those families have members who kept family tree records. Of course there are some Jews who carry the name Cohen, Kahn, Kohen, etc. who were not so meticilous so they would be considered as being "Cohen enough for the current day." Yet, if there was a rebuilt Temple in Jerusalem those would be the first elimanted from the choice of who is to serve in the Temple.
That being said, there are some Jewish families who have a similar situation and a claim to descend from King David. Last names may be Dayyan, Shereshevski, Ma'atuf Dohh, Shemakh, Ram, etc.
Also, Davidic descendents were still around during the 2nd Temple period and way after. There was a point in Jewish history where the leader of the Jewish people, post 2nd Temple, was always from the Davidic background.