A battery is a very large capacitor. A very large capacitor does not respond well or much at all to high frequency voltages - hence battery testers use low frequency AC voltages to measure the AC component of battery impedance.
A square wave contains only very high frequencies and zero-HZ flat DC portions. The very high frequencies are what's forming the nearly vertical leading and trailing edges of the square waves that show on an oscilloscope and ... the large capacitor/battery can't respond to, or "follow", these edges.
I'm not sure what an old lab tester is measuring. It's probably just providing a highly accurate value for a battery's DC resistance portion of it's total impedance via the form of Ohm's Law where DC resistance (R) = equals delta-voltage (change in voltage) divided by delta-current (resulting change in current). The applied voltage "delta" part comes from over and over application of a square wave's high portion and low portion.
The above is my best quess ... which is dangerous without drinking enough coffee before making a quess.
