I would like to point out the obvious here - it is not just fighter pilots that get nicknames, but they are perhaps the only group with an egotistical need for them.
OK - theirs are 'official' and become callsigns, but that is only because they need something to show the world how important they think they are and an attempt to justify their existence. Do Surgeons have callsigns? Do Nuclear watchkeepers? Do Prime Ministers and Presidents (maybe amongst the secret service). No. They do not. The reason is that they have genuinely busy and engaging jobs with daily value and don't need to try to justify their existence. However, 'Fighter Pilots' the vast majority of whom have never been in a fight, do have the biblically enormous ego of a 'Fighter Pilot' to pander to. They are self-actualizing narcissists.
Look at the RAF. Want to speak to guys who have seen action up close, been shot at, had rounds through the cockpit more than once, seen death up close?...Then go to Odiham or Benson and listen to the Support Helicopter guys. The warfighters in the RAF fly helicopters, and everyone knows it. Fast jet guys fly fast jets, not fighters, because they do not fight and have not done so for years. They have no idea what true combat is like - it is all ego and bravado...go to an Odiham crewroom and you will soon know what a 1000 yard stare looks like.
The sharp end of the RAF has rotor blades, not wings...and they don't need callsigns to show the world their worth...