Hardly; for one, frequently the Jews were in the host country long enough to share many ethnic traits with each other through some combination of adaptation, conversion, intermarriage, infidelity, and rape. For another, many people who identify with a national identity aren't originally from the country, or have a similar ethnicity to the prevailing majority - just look at second generation immigrants in many (though not all) Western countries.
I think the difference for a national identity is enfranchisement. Obviously, citizenship and equal rights are part of this, but I take an even broader view - if the individuals/group in question feel an integral and equal part of society where they are involved in building and directing the future of said society, they're a whole lot more likely to adopt the same national identity than if they're set apart intentionally from social, political, and legal perspectives.