I couldn't figure out a way to replace it with new one as the ringing button stops me from sliding it out and it seems there's no way to just remove it from other direction.
Could be that it was removed for painting and put back on before the paint was dry. Or someone used some silicone as well. Do you have a Hauswart? Or can you ask the landord? Or send a pic to the manufacturer and ask for instructions.
Will it slide if you press the button fully? Noisy but might work. Otherwise ask the Regie/Concierge to fix it. In meantime stick a piece of paper with your name over it.
Can you slide the thing upwards (or sideways, or rotate)?
That said, your new nameplate won't fit entirely anyway as it has no bellbutton hole. Whether width and thickness match is yet to be seen, too. A pragmatic alternative might be to to glue the new nameplate onto the old. Or maybe, if you're not sure you can live with it, start with a sticky strip.
It's a single family house and I'm the owner, assuming there's no concierge.
The door comes with an electronic switch and I'm thinking to replace the whole plate with a modern one (with video camera and networking). Maybe I should just try harder in that case?