I have heard quite a few times now aircraft screwing with ATC radar because there Mode S ident is wrong usually with an exam callsign on it.
True. When Mode-S became mandatory in NL there was an initial period when a load of airplanes flew around with wrong flight IDs. Either by accident or on purpose. "JESUS" was apparently the one that made the headlines most prominently.
There was some stern communication from the authorities to all owners/operators, followed by some stern communication from clubs/groups to their members not to mess with the settings (after having verified the settings were correct, of course), and the problem seems to have disappeared now.
What's an "exam callsign" by the way? Is it even allowed to put a flight ID on the transponder other than the aircrafts callsign (e.g. GABCD) or the flight number in case of CAT (e.g. KL8225)?
When asked to change it the pilot doesn't have a clue how to do it.
Depending on the transponder this is actually quite complicated. With the full-width transponder models from, for instance, Garmin, it's doable in-flight if you know what you're doing. But with, for instance, those Trig transponders that fit in an instrument hole, and only have three or four buttons in total it's a complete nightmare. In fact, it's already quite a challenge to put a squawk code (other than 7000) in those, while at the same time keeping a decent lookout.
I agree it's not right, but I don't think ATC should expect pilots to be able to change their flight ID in-flight.