It is no longer possible to book train tickets in Germany without providing your verified email or phone number. FYI, all phone numbers are linked to national ID cards in Germany.
https://www.thelocal.de/20230925/everything-that-changes-in-october-2023-in-germany
The difference is that a savings ticket is bound to a specific train, whereas the full-price tickets have free train choice on the day of the ticket.