It's very easy to check how accurate the ESR measurement is. Take a cap, measure the ESR, then put a resistor (preferably metal film), put it in series with the capacitor, measure again. The second measurement should be equal to the first + the resistance of the resistor.
I'm going out on a limb here, but seeing as the DE-5000 is actually a proper instrument, and the MESR-100 looks like it's been slapped together in a Chinese basement, I'd say the DE-5000 is the one that gives accurate readings here.
The DE-5000's accuracy is given in the manual for all the ranges. What's the accuracy spec for the MESR-100? Oh, that's right, there is none. It's essentially just a random number generator.
The DE-5000 has several frequency/capacitance ranges, like every other LCR meter. You can't measure large capacitances at high frequencies, because the impedance of the DUT would be too low and thus the resulting voltage would be too low to measure. And it would be pointless, as well. That's why capacitor datasheets specify the capacitance
at 120 Hz for electrolytics or 1kHz for ceramics or film caps.
Yes, the capacitance changes with frequency
depending on the type of cap. Electrolytics are extremely non linear. Try a a decent film cap, and you'll see that the capacitance will measure essentially the same from 100 Hz - 100 kHz.
An electrolytic cap that has 2200uF printed on it really means "this cap has 2200uF
at 120Hz". At any other frequency the capacitance will be different.
In any case, measuring capacitance of electrolytics is a pointless exercise, because, they're usually specced at -20+80%. What does measuring the capacitance tell you?
You can still measure the ESR of any DUT with the DE-5000. Simply switch to Rs mode. That will measure the real part of the impedance, i.e., ESR.
As Wytnucls already explained, measuring ESR to check if a cap is good or not isn't very helpful. Measuring DF is much more informative. Measuring ESR can be useful when you're comparing different types of caps for a certain application. Measuring ESR is a crutch, it's just better than nothing if you don't have a proper LCR meter.
So, the DE-5000 works fine and is very accurate. You just need to understand AC circuit theory better.