fyi, /u/OldWorldGlory rounded up a few links to earlier posts here
Oh but they did. yet, one must consider that by that time most European countries had extensive records of every individual legally living in their borders, in those records the nationality was also registered. add to that the fact some Jewish communities (especially in Poland) had some distinct features, like a common dress code, haircuts, family names and accents that set them apart from the local population. also, in some places you could count on the local population in identifying the Jews for you if records were unavailable.
basically there were Jews that successfully lied about their nationality and were able to escape to tell about that, but in most cases one couldn't fool the Nazis.