Whirlygig:
Not quite - ICAO controls callsign allocation, and you can send them a form with a choice of names and they'll see which is suitable. Bear in mind it's possible that callsigns on one aircraft might be heard in London on on day and Sydney the next, hence the global co-ordination.
I seem to remember it's free, too.
As an aside, I was pleasure flying at an airshow some years ago, and was asked to use the callsign "Jetranger" - made it easier for all the f/w folks to see and identify me. There were times when I was asked to stop flying - sometimes the display pilots were ok with me flying a deconflicted route away and back from the airfield, sometimes not. On about the fourth call for start, I asled for a callsign change to "Optimist 1" - got to go flying too !