I don't recall being asked for any identification when going to register my child's birth. I wouldn't have had any, I think I was alone and the registrar took the information down and that was it.(circa 1979).
1980 the Registrar visited the hospital to record births from the mothers in the maternity ward. I wasn't there but I am registered as the father.
Imagine when births happened at home with a midwife, if you were lucky, how easy it would be to register anybody as parents, maybe in a distant town. In history there were no passports, no driving licences, no voting register etc.