The labels look like this on our screens. There are no honking alarms, because the whole world (outside the US) is not covered by radar! We look at the symbol, and provide the service appropriate. We know our radar coverage and take action when a circle unexpectedly turns into a box.
The ABC123 on the left is being received by SSR & Primary radar, if the pilot turns off his transponder or it fails the label stays on the screen changes to the display on the right. We would notice and take action.
The GHI789 on the left is in SSR coverage only, if the pilot turns off his transponder my console displays the label on the right at the position it thinks it should be based on the flight plan and any position reports entered. The level displayed is the last reported by the pilot or transponder. The transition between the two is a bit more complicated than that, and we can activate a thing which puts a green starfish at the position we last had radar contact, but don't because all the starfish get annoying after a while...
This is for Australia, your ANSP my do something different. Those who designed the system that got the GOL should be f&^ked and burnt because they chose some funny things to display.