Update: I had visited the website of the manufacturer and filled in the contact form, with the question if I can maybe get a replacement pump. Here is the answer:
"Dear Mrs ...,
We regret that you have problems with our product.
For proper cleaning, it is advisable, depending on the water hardness, to place the pump in water and vinegar overnight, then rinse well, dry and finally lubricating the impeller (paying attention to suitable lubricant). Please keep in mind the pump never run dry.
In compliance with the proper care, there should be no short-term defects.
Should this advice not lead to success, please feel free to Amazon, which will be happy to provide further assistance, because it is still under warranty.
Yours sincerely ... "
Because our water is really very calcareous, I put the pump in water in which I have resolved the Entkalkungszeug for tea and inserted the plug. After some time, the pump actually ran again. In addition to regular cleaning, I have it now so regular descaling. Best not only the pump, but also the rest of the well. However, it is unclear how the plastic will respond to the frequent descaling. Unfortunately, I still do not know exactly what to do with the impeller (according to Wikipedia "a of a ring-shaped or tubular housing enclosed propeller"), because I would have to disassemble the pump for this purpose so only once. And what is a suitable lubricant for a propeller in a cat drinking fountain? Possibly problems will therefore still arise.
Because of the hot-werdendem PSU I dazwischengehängt a timer. He now runs ever for an hour for the time being and then has an hour break.
It remains to be seen what will happen next.