Every aircraft has an identifying number - for some reason 16 digits sticks in my mind but I am probably wrong. Think of it like the VIN number on your cars engine chassis. That means that the number identifies the aircraft type for life.
The callsign that ATCOs see on the radar are derived from the allocated squawk - i.e. they are paired up on an individual basis. These squawks have a shelf life so that if an aircraft is delayed for a while, a new squawk may need to be allocated. This means that the squawks can keep being used as there is a finite number.
The other info that ATCOs can see i.e. Mode S (as this is what you are asking) are downlinked from each aircraft i.e. IAS, HDG, Selected Flight Level, ROD/ROC etc. There are hundreds of downloadable items, but ATC only use a handful.