A "call sign" is a business' name plus the (typically) three or four digit flight number.
In all ICAO member countries, private pilots use the airplane's registration number when communicating with ATC. There isn't a flight number, nor is there a company name, so the plane's registration number is all that's available to identify the flight.
Similarly, the big commercial carriers always use call signs. The "business name" is typically registered with the government, but the numbers themselves are assigned by the airline in whatever system they want to use. Usually, the only rule is that you can't have two planes with the same call sign in same airspace simultaneously (although some countries do have stricter rules than that).
For charters, small commuter airlines, etc., there is a lot of variation between countries, so I can't get more specific without knowing which country you're asking about. For instance, the FAA allows any corporation that's registered as an airline and certain charity organizations to request to be allowed to use call signs, but it's not required; they can continue to use airplane registration numbers if they so choose.