How does ATC know with aircraft is which and how is the callsign transmitted to ATC?

Now, jet aircraft often have a Mode S transponder, so does that transmit the callsign directly from what callsign is entered in the FMC? And if that's the case then why is a squawk code required, because if I'm not wrong smaller aircraft without Mode S, transmits their callsign using the squawk assigned to them which makes sense, but how does it work for jet aircraft with Mode S?

Controllers will assign a flight or registration number to the discrete transponder code assigned to each aircraft under atc control. This assignment in their computer systems will allow the radar contact to be tagged with that id on the radar screen.