MOSFET unity gain frequency
I suspect it's because "unity" means unity, not some value with units attached. So we can talk about unity current gain or unity voltage gain independent of any choice of unit system. But if we say a transimpedance is "unity" that depends on our rather arbitrary choice of definition of the amp and volt.

And we can't talk about the voltage gain of a MOSFET in isolation, because (at least in the common-source configuration) determined partly by external components.

So the author chose to find the unity current gain frequency because that's a meaningful value.


  why should we care about this frequency?


It's a figure of merit that gives a rough idea of the maximum frequency where the device can provide useful gain in a circuit. 

If output impedance is higher than input impedance we could provide voltage gain above this frequency, but usually we want the opposite: low output impedance and high input impedance.