Well, considering the price of that Elektor meter, I would say that this is an epic fail from Elektor and a very poorly designed piece of kit. At the very least I would expect the USB interface to be isolated, otherwise the PC noise and ground loops are going to swamp whatever you are measuring - as you have discovered.
BTW, you should clean off the flux residue that is visible on the boards. I think that that board is just the display and there isn't anything analog on it, but there shouldn't be flux left on boards in this grade of instrument as a matter of course.
For the price I have seen this thing sold for you can have a decent brandname meter - e.g. one of those very popular Peaktech/Atlas analysers. The accuracy and resolution are comparable and no hacking is required. Or even a second hand Agilent, if you really need a very good quality meter.
I have got a cheap Chinese LCR for about $30 which is good enough for my radio stuff. Realistically, how often do you need to measure contact capacitance or use 0.1pF resolution? Seeing the construction and the sort of hacks you had to do, I would have difficulties trusting the accuracy of the measurement of that Elektor anyway - those parasitcs measurements are really fiddly and just looking at the clips wrong can easily produce more than 100% error.
Re switching PSU issues where you need to measure ESR - there you don't really need that much accuracy. Most often you need to only check that the ESR is within the expected range, i.e. that the cap didn't dry out thanks to high ripple current passing through it and its ESR is now too high for the circuit to function correctly.
There a specialised, two wire ESR meter is much more practical than a general purpose LCR meter, because you can measure the caps in-circuit. Again a $20-30 on eBay and you are set. The added advantage is that when (not if) you manage to "measure" a charged cap with it, you didn't blow up a $300+ instrument ...
BTW, the Mendel90 I have got from you is still going strong, thanks a lot for that!