It works just the same in Germany. People usually show up with their voter notification letter that has their name, address, date of birth and voter number on it. Unless 20 year old Janine shows up with the letter sent to 87 year old Hans, it is assumed that everything is in order. Her name will be crossed off the list and the letter of notification will be withheld, because it helps the volunteers with counting later on (think “We have 875 vote slips in the ballot box; do we also have the corresponding 875 voter notification letters that people left?”).

That said, people can also show up without the letter and present their ID card (Personalausweis) or passport. That isn’t strictly required, however. Both voter notification and official ID are fine on their own.

Edit: What is true, however, is that you cannot choose where to vote. There is only one voting location that has you on their list of eligibility.