Mount bathroom furniture on walls tiled halfway up

The preferred tool would probably be a router. It provides a clean edge and works very well with this material thickness.

However, of course only if the frame protrudes at the back of the cabinet. If the back panel is flush, then of course not.

Remember that a cabinet like this needs ventilation at the back. If you shorten the side panel, you also reduce the ventilation, in the worst case down to zero.
It’s better to double up the side panel and stick a plastic cover in front of it or something like that.
For acrylic, you have to be an absolute master so that a 10-12mm gap to bridge looks good afterward, even with joint filler profiles.

If the mounting no longer fits because you double up, there are spacers of 5, 10, 20, 30mm available at any hardware store. That’s how I screwed my cabinets rock-solid to the wall with spacing.

Did I understand the problem correctly?

That's how I understood it too. However, I don't see a problem there. Just hang the wall cabinets a bit higher so that they end exactly above the tiles.
We did the same with our bathroom mirror cabinet and it results in a perfect height. Or rather, we did it the other way around, first determined the height of the mirror cabinet and then tiled right up to it, which was supposed to be around 1.20 m, but ended up being 1.25 m. But 1.20 m would have worked wonderfully as well.

Unfortunately, I don't have a photo yet since the cabinets are not installed. However, the tiles are finished, and the wall is wallpapered and painted. Yes, it's about the gap, purely aesthetic. It just looks stupid. The back panel is recessed by about 2cm. This would allow notching out the side panel. Doubling sounds interesting, though. But the question is whether it will really look good afterwards.

The cabinets are 1.30m high. There's no way to hang them just above or just below. Unfortunately, I always catch both areas of the wall.

How high is the ceiling? If it is under 2.5 m, then those are simply the wrong cabinets. Are you handy? Then just cut it out. Otherwise, the professionals will do it.

Yes that's how I understood it as well and that's at least how the situation presents itself for us. Our cabinet is 175cm high, and we have no choice regarding whether to hang it higher or lower. In any case, I will not touch the cabinet, the edges are veneered all around and thus (hopefully) protected against moisture, otherwise I would have to come up with something else. Fortunately, it is not the main bathroom for us, so it will remain a minimal solution, alternatively the cabinet with spacing.