Does Elon Musk of Space X need a license to launch spaceships and starships? If so, what is that license called?

Yes, you need a launch permit to launch anything into the sky. The FAA must keep airplanes away from the area so they don't collide with your rocket. People must be kept away from the launch area. The military must know it is not a missle attack. 

Lol! A "license" to launch "spaceships" and "starships"! Sounds exactly the sort of things you would find in a Cracker Jack box, or on the back page of a comic along with advertisements for "X ray specs", money printing machines, or gadgets to turn yourself into a ventriloquist..

SDL aka Space Driver's License

For Elon Musk, I should think it's a Crash License.

Well... there's dozens to hundreds of permits required to build the structures he does, to ship/manufacture the fuels used as well as their storage, etc., which requires licensed contractors and operators.  

  To *launch* a rocket, he needs a clearance by the FAA for the airspace around a particular site - which includes down-range clearance, and a clearance by the coast guard for down-range safety if it occurs over the ocean.  There's also an announcement (I'm not sure how that's done officially) to other nations, alerting them to a launch occurring. (So they won't think it's an ICBM coming their way...)

I don't know if it is called a license as such but every launch has to get FAA approval.

No, the there is NO license required. Musk has LEGAL CO TRACTS with You really are totally clueless.