It therefore seems as if the shape of one measured signal depends to some extend on signals on other analog ports being measured.
It does (in some circumstances). There is only one ADC, shared between the analog pins. The result is that one reading can be impacted by the last. A common workaround is to read twice, ignoring the first reading. When I posted something like this recently, Grumpy Mike added this:
That is only true if the input impedance to the ADC is significantly higher than 10K.
Keep it at 10K and you can sample at the maximum rate.