In the US, you probably have to have a Social Security number to open an account with a mainstream cell phone provider.
To use "tracfone," though, you don't need a Social Security number.
I'm sure, in Europe, it's required to have some sort of identification number if you want to open an account with a mainstream cell phone provider.
But you can get "sim cards," and "pay as you go." To "pay as you go," you don't need identification.