The US doesn’t keep registries of religious buildings either. There’s obviously titles and whatnot for buildings. But it would likely be deemed a constitutional violation to require registration of every place that would be used to hold religious services. Some in Germany have proposed a “mosque registry”: https://www.dw.com/en/german-politician-wants-islam-law-and-....

A much more accurate analogy might be a registry for religious paraphernalia (crosses, rosaries, etc), including information about the purchaser. Such items are tied to an individual, and thus serve as a very effective proxy.