Quick update. I spoke with a local window installer. He wants me to send pictures first and then he will decide if he can do anything about the window, or he might come by and take a closer look. Apparently, every prefab house manufacturer has a slightly different system for sealing the windows. Attached is a picture of the leaky spot on the kitchen window. It looks similar on the opposite side.
Oh man... the house is about to collapse. And you want to open up the facade for that? Better put the money into an extra loan repayment.
These stains are not very strong at the moment. However, they change (depending on the intensity and duration of rainfall). When it rains for a long time, the stains can sometimes be as large as a one-euro coin and also on the exterior wall (so not just on the inside next to the window). Strangely enough, the stains also disappear very quickly again. I measured the area myself with a moisture meter (although it is a cheap one with two points at the front). It showed significantly higher values here than on the rest of the house. Where I measured, I had to tap into the plaster. There are now especially many stains here. Hopefully, I haven't damaged anything there.
My guess is, this is a case for a painter/ plasterer or the plaster manufacturer. Are the stains only in the window reveals? Maybe something is oxidizing with the plaster there, causing it to lose its water-repellent ability in those spots and thus become more damp?