Is it possible for ATC to see who’s transmitting on the radio on their radar?
A version of this has been implemented in some areas, such as the Maastricht Upper Area Control Centre, using radio direction finding equipment.
The EUROCONTROL website provides an explanation of the system, including an image of how it appears on radar displays:

Using triangulation software, the Radio Direction Finder, recently deployed throughout MUAC’s international airspace (the upper airspace of Belgium, the Netherlands, Luxembourg and north-west Germany) calculates accurately an aircraft’s position on the basis of its radio transmissions.
In densely occupied airspace, the Radio Direction Finder assists the controllers in rapidly identifying which aircraft is transmitting on the frequency.  This is particularly useful in MUAC’s sectors where controllers handle up to 25 aircraft at any one time, and is a prerequisite for the implementation of the free route airspace, planned at the end of the year. The origin of a voice transmission appears clearly on the controller’s integrated Human Machine Interface and is totally independent of conventional radar- and GPS-based aircraft localising techniques; this feature contributes to reducing call sign confusion, read-backs from wrong aircraft or crossed transmissions.