>>>Since aviation radios are full duplex, how can it distinguish between mic clicks and voice at the same time?
Aviation radios are a simplex type. They send and receive on the same frequency -and can only do one at a time. FYI, marine comm radios have duplex channels (20, 24-28, & 84-88) for communicating with land lines via a marine operator. Pilots do not have such a luxury.
Distinguishing mike clicks is actually quite easy. AvCom radios use something called double-sideband suppressed-carrier amplitude modulation to transmit voice. The receiving radio suppresses the carrier wave and only amps the voice wave, which is demodulated from the carrier. The PCL controller needs only to listen for pulses of the carrier wave on the CTAF frequency. This makes it a much simpler device than the radio in your aircraft. It doesn't even need to demodulate the signal to separate the voice transmission.