This is interesting (my emphasis) — Polyamorous throuple fight to keep their names on children’s birth certificates - NZ Herald:

A polyamorous throuple won the right to put all three of their names on their children’s birth certificates, but have been pulled back to court after the Attorney-General appealed the decision.

… the appeal wasn’t because of the family’s sexual orientation, but simply because it wasn’t possible to have more than two people listed in the system. …

Perkins said there was international consistency used in registering information from Births, Deaths, and Marriages, and allowing this throuple an exemption simply couldn’t work.

Seems to me there are places around the world where families differ significantly from two adults and one or more children. What about places where there are multiple spouses? Surely an internationally consistent system would have to allow for more options.