I had to use Zonal Statistics as Table (ZST), in order to extract data of a raster-file within some shapefiles. The trouble was that some of the shapefiles were overlapping, which ZST can't work with, i.e. the shapefiles are not allowed to be overlapping if ZST are to work proberly.
This problem can be evaded if ZST got a script implemented, that run ZST for each FID individually and then assembles the results in one table afterwards.
So the main idea is, that instead of running ZST for all shapefiles at the same time, ZST should run for each FID individually.