In the air traffic control system, call signs (especially for commercial airliners) are temporarily assigned based on function (the flight it's on). And gives a bit of information to the controllers about the flight. Since there's already a lot of procedures in place for call sign management, this is just one additional piece.
The whole history of the President's plane being assigned a special sign is due to an incident where the plane's call sign conflicted with that of another flight.
There is no space traffic control center (certainly not one that is using voice communication for real-time interaction), and there is no procedural system in place that assigns and uses call signs to deconflict communications. Piloted launch vehicles do not communicate with ATC or use ATC callsigns during the mission. It is still a rare occurrence today for communication to take place with multiple piloted space vehicles, and never is this happening without lots of planning.
- There would be no operational benefit today for mission communications to use a special identifier.
- I'm sure someone, somewhere during the planning would think it a cute idea and suggest it. So it could happen, but it wouldn't be to comply with any existing policy.
Someday in the future where the president's spacecraft might show up without someone expecting it, and if we're using voice communications to do spacecraft traffic control, this might be a good idea.