This has been answered elsewhere (linked below), but only partially, so I thought I'd add a bit more.

In addition to individual registration by religion, the Nazis demanded proof of non-Jewish ancestry. That meant that, while you might be a converted Catholic or Protestant and registered as such, but if your parents or grandparents were born Jewish, then you could still have problems.

An easy way to confirm this was their registrations, but also the church practice of recording life events. If you couldn't produce baptism, confirmation and marriage documents from a church for every grandparent and parent (and further back for some positions), you could still be considered Jewish enough to be persecuted, even if you personally were registered as a Christian.

In smaller towns and villages, being known to be Jewish or of Jewish heritage by other locals could also be enough, particularly in occupied territories. Belarusian Jewish villagers weren't given a chance to produce documents prior to being herded en masse to mass shooting sites, for example.

There was one weird option for a small number of people however, particularly those who happened to look more of what the Nazis thought an aryan should look like. In some cases, one could try and apply to be "aryanized." This involved some falsification of history, such as a sworn statement that a woman had cheated on her Jewish husband with an Aryan "real" father of her children. The person in question would then be examined by "specialists" who would look for what they considered the physical markers of an aryan. This was really not an option for most people though, and even less so in occupied territories.

fyi, there have been a few threads on this question, so while not discouraging more discussion here, you may be interested in getting started on these

• How were Jews identified by Nazis? What happens if a Jew didn't wear the Star of David while walking on the streets in a Nazi-occupied town/city? - featuring /u/commiespaceinvader

and, /u/OldWorldGlory rounded up a few links to earlier posts here

• How did nazis identify who was Jewish or not?

In the USSR, all identification documents included information about one's 'nationality', a category which more closely matches our english-language concept of 'ethnicity'. 'Jewish' was one of the recognized nationalities of the Soviet Union, so all ID's or lists of apartment residents would have made it clear that a citizen was Jewish. Therefore, even in places where religion and religious practice were not a factor, one's Jewish ancestry could still be identified. As for why people would not simply hide their IDs, in occupied territories the Germans put in place rules that basically assumed anyone without an ID had gotten rid of it to hide their racial status, which made it very suspicious to be a person with no ID. It was possible to get a new one that misrepresented your nationality/ethnicity/race, but it required bringing neighbours who would swear to who you were. Such a system opened up a dilemma for people trying to hide their identity. If you 'lose' your ID and leave your home, you will not be able to get a replacement and you will look suspicious. If you 'lose' your ID and stay where people know you, you can maybe get an inaccurate new one, but could also be denounced by neighbours who knew you were Jewish. The role of non-Jewish neighbours in the identification or denunciation of Jews was a big factor. Also, many primary source accounts mention another way that Jewish men were identified--circumcision. Men who were suspected of being Jewish were often forced to drop their pants. If they were circumcised, they were outed as Jewish (unless they could convince the German that they were Muslim, as a few Prisoners of War apparently did). So there were many ways, beyond self-identification or religious practice, that Jews (that is, people the Germans considered Jewish, since they disregarded conversions) were identified.