Never used one of those but I have the impression what they measure (for transistors) is the gain, hFE. But for most matched transistor usage in synth circuits, according to Fritz, that isn’t what you want. What needs to be measured is base-emitter voltage, V_BE. Actually you’d like to measure leakage current but that’s hard and V_BE is a good enough proxy.
The CGS method evidently tries to measure that directly. The Fritz method, in his words: “Since you want to know the difference between two transistors, why not set up a circuit to measure that difference directly, rather than having to make two separate measurements and then subtract the results?” The point is that if you want to match them at the 50 µV level then with the CGS method or any method that just measures one transistor at a time you’d have to measure V_BE at a precision of about 1 part in 10^4 which is beyond the abilities of most inexpensive multimeters. With the Fritz method you’re just measuring the difference so a 10% or so measurement is fine.
Then again, for synth DIY you probably don’t need anywhere near 50 µV. I’ve read a match at 2 mV is good enough, that’s a 0.3% measurement which is manageable.