Earlier quoted context omitted.
There is not such thing as "Jewish" DNA. Is a culture and religion, but not a fully different race. Some genes could be in the past more represented, but it was just "Mediterranean dotation". A mix of European, African and Asian. Today is much more mixed fortunately.
If you can't prove family connection with paperwork, you can emigrate to Israel by means of DNA test which proves to the government that you are Jewish. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5034383/ Couldn't find a page on the government website, but I remember reading about it there in the not too distant past.
A DNA test can be used to obtain Israeli citizenship, but this is reserved to prove that a person is the child of an Israeli citizen. According to Israeli law, if a child is born to an Israeli mother or father abroad, they can be granted Israeli citizenship. The DNA test is used to authorize this, proving this familial link. We discuss more about obtaining a paternity test in Israel in another article.
In some very rare cases, a DNA test can be used to prove their relation to a Jewish parent, sibling or grandparent, even though the applicant doesn’t have documents proving this relationship, but said person has to have documents verifying that they are Jewish.
https://lawoffice.org.il/en/israeli-citizenship-dna-test/