The Berlin City library (ZLB) is different than the State Library (Staatsbibliothek zu Berlin) that you mentioned. The English pages of both should give a general idea of the conditions of most public libraries. For the Staatsbibliothek, fees have been abolished for the next 2 years, but registration is still needed.
As a general rule, if a book does not need to be retrieved from the Magazin, you don't need a readers card.
Registration - Zentral- und Landesbibliothek Berlin
While you are able to visit the ZLB and use many of its services without a library card, you will need one if you want to take materials home and/or use our advanced services. To get a card, it’s best to go directly to the ZLB or use one of Berlin’s other public libraries.
Berlin-Sammlungen | Senatsbibliothek Berlin
Breite Str. 30-36, 10178 Berlin-Mitte
Here I know that in the main hall newspapers and magazines are available that can be freely selected and read at nearby chairs.
There are also many rooms, depending on the topic, where a shelf book can be read in the room.
I have only been asked once for the card when reserving a micro film machine for the newspaper archives.
The city libraries are often specialized (the above is for Berlin history) but all use only 1 card system.
The Frankfurt site does not seem to offer an English version, but the basic offer seems to be the same as in Berlin.
Frankfurt am Main: Zentralbibliothek
Stadtbücherei Frankfurt am Main
60311 Frankfurt am Main
Mo, Di, Mi, Do, Fr 10-19 / Sa 11-17 Uhr
Media in other languages: English, French, Russian, Spanish, Turkish
Magazines and newspapers German and International.
Hinweis: Der Besuch der Bibliothek und die Nutzung der Bestände vor Ort sind grundsätzlich kostenfrei.
Note: The visit to the library and the use of stocks on site are always free.
This is only one of the 19 public libraries in Frankfurt. So here too, you must choose one that fits your needs.