<p>Suddenly, my printer has started producing prints that have a very pronounced layering. Normally, the alignment between layers is very good, and the prints look very smooth. Suddenly, the prints have become much worse and the layers are misaligned with respect to each other.</p>

<p><a href="https://i.stack.imgur.com/MgWVx.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/MgWVx.png" alt="enter image description here"></a></p>

<p>The part on the left is my "normal" quality, while the part on the right show the deterioration. Here is another picture (in which the good part is on the right):</p>

<p><a href="https://i.stack.imgur.com/c1I5Q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/c1I5Q.png" alt="enter image description here"></a></p>

<p>The parts are both printed with 0.1mm layer height, and identical slicer settings/filament. I am printing on a custom-built FDM printer; the mechanism is roughly similar to that of an Ultimaker.</p>

There are many factors, here are a few things to check:
I'd first suspect filament feeding.  This type of ridging can be caused by a filament coil that is binding occasionally, or a filament that doesn't have an even diameter or volume per length.  Binding within the filament feeder and feeder tubes can also be a cause. Bubbles in the filament, or sometimes a mismatch between the filament ideal temperature and the head temperature could create results like this, but it probably wouldn't vary so much between the layers.
Next I'd look at the print head.  If it has blockages, or poor temperature control this could result.
Lastly, I'd check the mechanisms - disconnect the motors and see if all the carriages slide smoothly without any binding, particularly the Z axis.  It doesn't look like you're missing steps, but binding here may result in greater backlash, which could result in similar ridges. Make sure any belts and gears are tight.