I found some information in Life and Death in the Third Reich (2009) by Peter Fritzsche. According to the author, blank forms were purchased in bookshops and filled out, with things like baptismal certificates, insurance cards, divorce records. Then it would be notarized by a church official or a civil registrar. So I don't think there was necessarily an official register with copies.
(There are also cases of officials 'cooking the books' for people who were "genetic undesirables" and giving them an official, fake, genealogy)
Even wilder, just having an Ahnenpass wasn't the be-all, end-all: people were also required to get certificates of genetic health, and those found to be 'deficient' could be sterilized, whether they were Aryan or not.