Thank you for your contributions. I'll try to respond here.
- yes, it is an old building, moisture is mostly on the edges, towards the wall. However, the dark spots can also be found in the middle.
- the grout was mixed as well as possible according to the packaging instructions. However, I had to add water because it was way too stiff. But if that was the cause, it would have to be equally dark everywhere. I only mixed for the whole room twice, so it can't be due to different mixtures.
- tile adhesive was pressed up between the tiles in some places. I wanted to scrape it out at first, but after I realized how hard it was and that the edges of the tiles were already slightly damaged, I left everything as it was. My thought was, put grout on it and it won't be visible anymore.