Porbbaly not to do with "time stamping" as that is superimposed on the recording machines well away from the transmitters... (Unless some countries have a different scheme?). It takes the form of morse characters to indicate each minute.
The clicking noise is likely to be is the controller using a handset or hand mic rather than a headset. Some ATC facilities have telephone type handsets equipped with a press-to-talk switch on the handle. This definitely produces clicks during transmissions. Controllers hear the same noises when pilots use hand mics.
If the controller is using a hand mic/ handset and a loudspeaker the "beep" is likely to be a very short burst of feedback when the transmitter comes on before the receiver is muted. We used to hear that on BAC 1-11s
Now somebody will say I'm talking a load of ****!!