It's Denmark, it won't have 1k babies born the same day.
It's the same in Sweden: YYYY-MM-DD-XXXX is the format for a personnummer, double the population of Denmark and there are no collisions.
Immigrants who don't know their birthdate are assigned xxxx-01-01, so they have actually run out of numbers for January 1st in some years.
That format looks like it would allow for up to 10k per day. Unless one of those X's is a check digit?
There have been collisions in Sweden, but that was due to lots of immigrants coming from the middle east that did not know when they where born. So many of them picked first of January. They “solved” it by overflowing to the next days.