I believe the metering circuit does not employ the voltage-independent "bridge" design. This renders reading voltage-dependent. When that feature is present one can get away with a wider range of voltages.
So as Matt says, using the wrong battery, while it will operate the meter, usually gives incorrect readings. Except for that brief moment when the battery decay passes through the 1.35-volt mark, which is what the nominal voltage of the now discontinued PX625 mercury oxide cells generated.
Adjusting the ISO setting to compensate is, as mentioned, a fiddly but usually acceptable shorter-term workaround. The problem arises because the discharge curves for the different replacement cells are not the same as the original mercury cells. This means if accuracy is desired, one must relatively frequently recalibrate (readjust) the ISO setting to follow the decaying replacement cell. Mercury cells held a very constant voltage over their entire working lifespan.
When the correct voltage is supplied, the meter is surprisingly accurate in normally lighted situations. I am an original owner of a Mat 124G and have a lifetime stash of the original mercury cells in the freezer. I once asked my camera repairman if he would officially calibrate the meter. He was hesitant as they are not particularly robust, but I insisted. After he finished I checked it against my handheld meters (again, in normal lighting situations, not in excessively dim light) and it matched exactly.
Years later it still matches. Note that I am not a heavy user of this meter. But when I have used it, it continues to read correctly.
Ken