The Berlin key ("Berliner Schlüssel") was mostly used in tenements with multiple backyard houses with a common front gate that should be closed at certain times. These kind of arrangements were mostly used in Berlin, hence the name.

The other German names for the key describe different aspects of it: "Durchsteckschlüssel" translates to "push-through-key" and "Doppelschlüssel" translates to "double-key".

If you are more curious about the thing, there is a book by philosopher Bruno Latour on the Berlin Key where he describes the thing with the sociological situation that lead to it if I remember correctly (was a few years ago when I read it).

Edit: The basic idea is that in this kind of multi-party environment they wanted to make sure the door is always locked in the evening (when people return). To unlock the door you have to push the key through. Once you are inside you can get the key back, but only if you lock the door behind you. So once the door is locked, it stays locked, unless someone from the inside leaves it open on purpose. This way people cannot "forget" to leave it open.