Mold problem in new kitchen - how to prevent it in the future?

Well, what can you do. Maybe wedge mineral wool between the refrigerator and the wall, nothing else/more will fit in there anyway. A freestanding refrigerator might also be an improvement. Definitely keep an eye on it though.

...or plan a new kitchen or redesign the old kitchen.

... which was just newly acquired, or did I misunderstand that?

Yes, this was just purchased recently. A new kitchen is out of the question – besides, as mentioned, there is no other possible location for a tall cabinet. Then rather a new refrigerator. 55cm freestanding unit instead of 60cm built-in, so that there is some space next to it. But I would also like to avoid that, since everything was newly purchased. I had also briefly considered guckuck2's suggestion about insulating the gap next to the cabinet – but then I thought that would actually prevent air circulation even more... or am I wrong to think that?

How do you come to that statement, or where can one find this information? I've never heard that before.

I picked it up somewhere. You can also find something about it if you google it accordingly.

I could already imagine the reason for this: A refrigerator emits heat, an exterior wall can be especially cold. A large temperature difference between the appliance and the wall results in condensation—just like with me. Hence my question about what could possibly be done to the wall here to prevent this in the future.
Regardless of whether one should do this or not—there was no other option in our kitchen, see picture. On the opposite side there are two doors (through rooms). There is no other usable location for such a tall cabinet.

I think you have captured the reason quite well. If I were you, I wouldn’t fiddle around with it, but eliminate the cause once and for all (we did the same with our problem corner in the bedroom, after fiddling around—except costing money—didn’t help). From my point of view, that would be insulating the exterior walls in the affected corner with climate boards. They come in 3cm or 5cm thickness. Depending on the temperature of the wall, it must be the narrow or the thick climate board. Usually, the thin one (3cm) suffices. That then needs to be plastered over. I would estimate about 4cm in total. Ergo: climate boards on both exterior walls and replace the built-in fridge with a freestanding one.

If you want to be absolutely sure: bring in a renovation company and ask for a free cost estimate. Then an expert will come with measuring devices and you will finally know the cause and can act accordingly.

OK, the insulation of the exterior walls is of course a REALLY big can of worms that you open there. I mean, how far do you want to push this exterior insulation, you don’t just insulate the exterior walls of this one room. The idea ends somewhere in insulating EVERYTHING all around. Then all of that has to be plastered or made nice in some way. That comes to some five-figure amount, which honestly I don’t just have lying around. Therefore, I am looking for a temporary quick fix solution ;)