In the US it was very common. My kids did it at school in the 90s.

The point was that if your child goes missing, you have their fingerprints available so they can be used in investigations. In our case, the fingerprints were given to us to take home in case they were ever needed. At the time I didn't think to ask if the police kept a record for themselves, but I doubt it (just because local PD wasn't very organized).