MacBook main volume can't be mounted during boot
Solution 1:
To conclude so far: SATA cable was not damaged, as suggested somewhere else.
I came to the conclusion that the SSD itself is broken and I failed to repair it. Even GParted was also not able to format the partition.
Therefore I have just replaced the SSD with a new one. Luckily I was able to save most of the data, since the partition could still be accessed from a Linux system.