why is the product of gain and bandwidth constant for an amplifier?
It only holds true for amplifiers whose bandwidth is limited by a 1st order capacitance such as Miller Capacitance on an BJT or the internal C compensation  for an Op Amp.  But for many types of Video and RF amps these configurations are not applied and hence are the same rules do not apply.

If a 1st low pass filter reduces -6dB per octave as found in the Miller intrinsic capacitance or the internal compensation of an OP Amp , we know -6dB means 1/2 Voltage and octave means x2 frequency so 1/2V x 2f = 1 for a normalized constant gain bandwidth.

Substitute the baseline gain and break point frequency or take the GBW product in order to determine 1 variable by choosing the other.