Here’s the simple way of understanding the system, I’ll use two examples, the first is my Foxbat.
It has an individual 24 bit/ 6 digit hex ID & the flight ID has been locked in as R1234 as an example as this combination will never change,(as per the Air Services requirements,) unless I decide to get a new mode S transponder or change my rego, on domestic flight plans my call sign is e.g. 1234, however my ATC call sign is Foxbat1234 ref AIP GEN 3.4-22
The next example I will use is an airline flight, again the 24 bit/hex code is an a/c’s individual ID, I am flying from Hong Kong-Taipie so when I set up the flight management system I must ensure that the ICAO airline code & flight number up to 7 digits long matches the call sign on the flight plan e.g. HDA486, however the ATC call sign is Dragon486, clear as mud?
Now if that original a/c goes tech & a swap is required, no problem the FMS is loaded on the new a/c, the a/c again has its individual hex code & HDA486 would be inserted as above.
When ATC look at their scopes they will see that R1234 is a recreational a/c for quick ID & QF1234 is a Qantas jet, have a look at Flight radar 24.