It is the formal registration of the birth, which is obligatory. Because it is about a Prince and Princess, it was all done with formality. The same happens in some other monarchies. There are also monarchies which make no any spectacle of a birth.
An example: HRH The Prince of Orange registers his second daughter, Princess Alexia before the Lord Mayor of The Hague:
http://www.refdag.nl/polopoly_fs/de..._marcela_laurentien_1_8126!image/63398015.jpg and for the ceremony one of the stylish 17th C salons of the Old Townhall was used.
It is just a more ceremonial, 'dressed up', form of the obligatory registration which every father has to do in most countries in Europe. A "normal father" registering a child simply does that at the public office of a townhall, just like going to a bank or so...