In the pre-
transistor era, all radios (and
computers!) were constructed with
vacuum tubes: Simply put, a negatively
charged cathode(-) when heated, emits
electrons which travelled to the
anode(+) thru a
grid(-) that regulated the flow--a
triode.
Battery-powered radios had a power supply made of three different types of batteries: Battery 'A' provided low voltage, high current power for the heating elements inside the vacuum tubes, Battery 'B' provided the power between cathodes and anodes and battery 'C' powered the grid circuitry.
Thus, everybody simply adopted the convention that 'A' battery referred to ones suitable to provide the glow current for vacuum tube heating elements. In this case, it is not a form factor like today's AA,AAA,C,D but voltage and mAh specs.