Transponder pulse signal width measurement
Answering your questions in reverse order.


  So, how do we identify if it is one kind of interrogation or another?


The answer is at the top of the page


"The SSR interrogation format ... consists of two pulses (P1 and P3) of 0.8 µs width which are separated by a certain time - this determines the mode of interrogation."

ie Two pulses are transmitted.  

Each pulse is of 0.8 uS duration. 

The time between the end of the first pulse and the start of the second pulse tells the receiver what response mode is required.


Duration between pulse edges in uS
 .......Mode
 3 ... Military 1
 5 ... Military 2
 8 ... Military 3 or Civilian - most common mode
 17 .. Unused
 21 .. Civilian Pressure &amp; Altitude query
 25 .. Never used.




  How is this measurement done in the circuit level?
  Can this be done by an ADC or microcontroller
  or is it done by software?


It would be done by any suitable means that the user decided to use.
 The pulse separation measurement could easily be done with hardware. 
 It could be done with software - and so a PC or dedicated computer system which may use a microcontroller may be used.  

BUT an ADC = Analog Digital Converter is used for converting analog signals to digital (as the name suggests). An ADC is not usually used for time mearurment.