Another officer can be glimpsed wearing the uniform at Starfleet HQ:
That may be the costume with captain's stripes seen here (scroll down)...
... though that one seems to have incorrect admiral shoulder marks.
I've always figured that it was a dress uniform authorized for all officers, and Kirk happened to be coming from a more formal setting at the time.
Robert Comsol wrote:
^^ That would explain the absence of the Starfleet insignia on the chest. The TOS dress uniforms didn't have one either.
Except for the court officer and Lt. Shaw in "Court Martial."