CA, then I did add to your post.
ZD, you are confusing the Communists and the Nazis. In the former Soviet Union a person’s id card included his nationality. The children of a mixed marriage were given their father’s (so Ivanov might be halachically Jewish whereas Rabinowitz is not). To add to the problem it was often possible to change one’s nationality and it was very advantageous to become a Russian. However, they did not right a “J” (in fat, the letter does not exist in the Cyrillic alphabet). They wrote “Yevrei” (Jewish), Russky (Russian), etc.
Changing one’s background also existed in Nazi Germany. A non-Jewish woman could testify in court that her Jewish husband was not the father of her child and make him an Aryan. Money and connections helped.
In any case, the problem will probably go away as the olim, especially those who are not Jewish, have formed a separate subgroup (I even hear children who were obviously born and educated here speaking to each other in Russian unlike children of other groups who generally speak in Hebrew among themselves. In addition, Russian families have a very low birth rate. Two kids is considered a large family. Many are also leaving for the US and Canada.