He doesn't know what he is. His identity is lost. You pointed that out yourself.
The identity of the individual tribes has been lost, but the knowledge of who is a Jew and who is a Gentile still exists.
Gentiles can trace their lineage and it isn't to the 'Lost Tribes', who are suppose to have made a migration (never recorded) after overthrowing the Assyrians (never recorded).
One thing that needs to be corrected. I never said that Israel overthrew the Assyrian empire, I just said they and Medes whipped the Assyrains hold them captive.
NonJews can trace their lineage to 721 BC? Really?