Exactly my thought. Every European country I've lived in works in pretty much the same way. Of course saying it comes from Japan makes it look more sophisticated ;)
Actually no, the koseki system or similar can be found in other Asian countries like South Korea but not in Europe.
In France you can get a birth certificate, a marriage certificate, and you have a "livret de famille" that regroups all these information but that's a document that you own, not the administration. There is no registry that regroups all the information about a given family.
Also note that in Japan, a women getting married will be transferred from her parents' koseki to her husband's koseki. There is no such thing in Europe.
European countries don't have a family-centered system like the Koseki (or Chinese Hukou; Wikipedia says South Korea abolished their system a decade ago) where a document represents a family.
But several of them do have a population register (which is individual-centered, but with references up and down in the family tree). In Sweden you don't get birth or marriage certificates. If you want to prove your birth or marriage, you get your population registry extract which shows that you are registered as existing (proves birth) or married.
So they're the same in that they are based on current state rather than collecting event documents. But they're different in that one represents families, whereas the other represents individuals.
Japan also has a separate residency register. While the koseki shows who is in a family (birth/parents, marriage/children), the residency register shows where you live now/before (and is used for taxation). European countries with a population register simply use the same registry for both purposes.
