I am not Jewish but I read somewhere that you have to provide paper to show your Jewish or that in some cases they take blood/dna tests. You would have to ask someone who knows more about it.
The criteria for becoming an Israeli citizen are not the same as the religious definition of Jewishness according to Jewish law.
According to Jewish law, a person who is born to a Jewish mother is a Jew (whether they keep the religion or not). A non-Jew who converts to the Jewish religion officially before a Jewish religious court becomes a Jew. If she's a woman, her children born after that will automatically be Jews.
Israeli civil law has its own rules of citizenship: A person born to a Jewish mother or father, or someone whose spouse is a Jew can become an Israeli citizen. That doesn't mean this person is recognized as a Jew, just that they have Israeli citizenship (there are many non-Jews who have Israeli citizenship). If a person converted to Judaism before coming to Israel, the Interior ministry checks if the conversion was valid according to their rules and if it is, they can become a citizen just like any other Jew. I never heard of DNA tests being involved in any way.