There are a number of issues for Jews just “blending in”, which broadly speaking worked differently in Eastern and Western Europe.
In Western Europe, maltreatment of Jews was mostly incremental. The Nazis didn’t start mass murdering people right out of the gate (though it was clear that things were going to get bad for the Jews, and people did openly speak in the ’30s of the possibility of Jews being killed en masse, albeit not on the scale it actually occurred). First, citizenship was revoked, certain rights were rolled back, Jews were de-integrated from society, etc. Eventually you got things like the famous yellow stars. The penalties for pretending not to be Jewish were pretty steep, and being killed for being Jewish wasn’t an immediate threat. Additionally, records were used over time to make it difficult to hide Jewish lineage. While Jews “passing” would’ve been relatively easy in Germany, the environment pre-war made it an unlikely choice.
In Eastern Europe, the Germans essentially rolled in followed by killing squads and created ghettos in cities. However, there were a couple of issues. First, you did have some of the same effect as in Western Europe, where people feared the Germans finding them unregistered more than they feared what’d happen if they didn’t. Creating ghettos and killing everybody wasn’t announced.
In Kiev, for instance, Jews were told that they had to assemble with their possessions for resettlement a few days after the occupation began–anyone violating the order would be shot. But upon their arrival, they were all shot in a ravine at Babi Yar–only a few who managed to slip away survived. While some Jews might’ve feared what would happen, the prospect of being hunted down was present, and culturally speaking restrictions on Jewish residence in Europe wasn’t exactly a new phenomenon.
Additionally, in Eastern Europe much more than western, Jews weren’t very integrated. It’d be a dead giveaway if someone could only speak Yiddish (which would’ve happened in rural areas, though most people could probably converse in the local language) or couldn’t find non-Jews to vouch for them or couldn’t document a name that wasn’t Jewish.
In all these cases, the penalty for trying to “pass” was death, and the Nazis had whatever historical records were available to hunt down those who tried.
However, some people did. While it was difficult to do it, among the millions of Jews some were bound to try it, and some succeeded. Just yesterday I heard someone asking about documenting Jewish genealogies because their Litvak (Lithuanian-Jewish) ancestors intentionally obfuscated their Jewish heritage. And I know a woman (this anecdote is illustrative, not a source–that’d be against the rules) who survived the war as a girl in a Belgian orphanage–it was assured that there was no documentation tying her to being Jewish, and by all appearances she was just a young girl who was abandoned.
A better-known example is the French village of Le Chambon-sur-Lignon, which sheltered thousands of French Jews (and others escaping Nazi persecution). Note that France had an integrated Jewish community (so people could “pass”) and didn’t have the incremental persecution increases in Germany, so there was more incentive to try to avoid the Germans figuring out you were Jewish at all. It required a organized effort, and lots of forged documents–if you mysteriously had no record of birth in the local church, the Nazis wouldn’t just shrug and move on.
So in short, you’d have to decide way in advance to pass in Germany, and it’d be difficult to in Eastern Europe. In either case, the incentive to hide being Jewish wasn’t immediately apparent, but the risks of it were.