More insidiously, the Nazi leaders approached the problem from the other direction: you had to prove you were not a Jew (or of Jewish ancestry). While registries of Jews (and others, asocials, 'gypsies', homosexuals, etc.) were kept, the main default was that, if you did not or could not "prove" your Germanness, you were assumed to be non-German.

After April 1933, for instance, proof of "Aryan" ancestry was required to acquire or hold onto all public-sector jobs. (This was per the "Law for the Restoration of the Professional Civil Service", initially designed to get rid of Social Democrats, but expanded to now exclude "non-Germans", where Germanness was not citizenship, but rather German/"Aryan" ancestry.)

So how do you "prove" that you are of "Aryan" descent? You have to provide all four grandparents' birth/baptism records (kept by the Protestant/Lutheran or Catholic churches...) Jews would not, of course, have baptismal records or birth-records in a protestant or catholic church.

As the years went on, laws requiring "Aryan" certification expanded to include everything from taking pharmacy exams to receiving unemployment benefits to (after the Nuremberg Laws) being able to marry.
The classic work on this is Burleigh and Wippermann's The Racial State, although I don't recall

As proof of Aryan (i.e. German but not Jewish) ancestry became a legal requirement for so many different things (including, ultimately, the right to sit on a park bench), the Ahnenpaß emerged. It was the "Ancestry Passport"... a red booklet that looked like a passport where you had the various authorities (again, primarily church baptismal records) sign off on your "Germannness."

You can still find many of these for sale to collectors.

An entire state ancestry office (the Reichsippenamt) was eventually set up to research and police claims to German ancestry.

I'm in mobile so can't type too much, but the answer is yes, they did eventually implement a registry of sorts. It wasn't a specific registry performed for Jewish persons, but rather that information could be found in the census, as it recorded a person's confession. They also kept registries of organizations such as Jewish associations See this USHMM article for a bit more detail.

They also did this in occupied territories. For example, one of the first move the Nazis did in the Netherlands was force people to sign a so-called Aryan Declaration, which by necessity excluded Jews. Combining this with regular census data gave the occupiers an easy method of identifying Jews in the country.

As I said, I'm on mobile, but am happy to go into more detail in a few hours after I get out of my next meeting.

Sorry for bad english, I need to translate from Dutch to English and it's prety late.

As far as complementing the other comments, the Dutch were neutral, but they had to do what the germans said. So as soon as the Germans got the power here, everything that happened in Germany, happened here as well. As far as the Jewish went.

But the Dutch were different. They already had a database of all people in the Netherlands, and so Jewish people as well. That made it very easy to find out who were jewish and who weren't. At the end of the were, nearly 75% of all jewish people in the Netherlands were murdered.

I know it's not what you asked for, but it's just my 2 cents.