This is commonly called Number Portability. I don't think it's mandated by the EU, but every EU country I looked up supported it. In the Netherlands, for examp

This is commonly called Number Portability. I don't think it's mandated by the EU, but every EU country I looked up supported it. In the Netherlands, for example, you can obtain a new SIM card or port number to another NL carrier. KPN uses the PUK code to verify ownership, which I think is great and has the least amount of hassle if you have written down the PUK code that was printed on the SIM package when you first bought it. The downside is that you cannot change the PUK codes.