https://worldpopulationreview.com/country-rankings/countries...
I don't exactly know what you mean by citizenship register but I can't imagine it's hard to workout who is a citizen and who isn't.
It’s therefore a lot harder to prove citizenship for an initial passport application in certain circumstances than you might expect. You need to prove that you have an unbroken link of people born in the UK to someone born before 1983, and as time goes on that will mean even more generations. Right now you typically need to provide your birth certificate, up to 2x parents birth certificate, and up to 4x grandparents birth certificates.
In many other countries the birth certificate will have the person numbers of the parents, which will mean there’s essentially guaranteed to be a record of the citizenship of the parents that the state can check. Alternatively there’s a national ID scheme that helps bootstrap this information early in life.
Anchoring proof of citizenship is going to become a very obnoxious problem going forward if there is not a population register or universal ID system introduced, as you'll have to go back however many generations it takes to reach birth before 1983.
I think the UK and Ireland are the only countries in the entire world that have non-birthright citizenship and no citizenship register, which is a less than ideal combination.