Indeed. I have read somewhere that until the 19th century, that moment was often months after birth. It was common to give the newborn away to a nurse if you could afford it until it had survived the first year, and only then accept it as part of the family. This way, the parents could safe themselves a bit from the excruciating pain of child loss.

Edit: This is e.g. mentioned in the German Wikipedia on Pox (https://de.wikipedia.org/wiki/Pocken#16._bis_19._Jahrhundert), there is also a citation for it. I have read it several times during the last years, but don't have a better source, sorry.