CAA will assign 'private' call-signs according to the protocols already >mentioned, in the manner of personalised number plates on cars, I guess.
Callsigns are used, instead of aircraft registrations, in situations where it will aid operations or billing (you knew money had to be there somewhere).
For airlines and ATC it helps that a callsign and number tells them where the flight is going. For other operations the callsign tells the authorities where to send the airways bill - normal practice is to bill the owner of the registration but these days many operators are lessees or managers and callsigns (or ICAO identifiers) speed the billing process.