From what he told me, there were some guitar models that didn't get a name tag until the final stages of production. You didn't know ahead of time whether a guitar was going to end up a Gibson or Epiphone.
I can't imagine that story you heard was about the electric guitars an basses.
When you start building an electric guitar or bass, one of the first things you do is define the shape it's going to be. So Gibson/Epiphone would have to choose which templates for the body and headstock they were going to use right at the beginning.
Most Epiphones had totally different body shapes and headstock shapes than their Gibson brethren.
I guess it could be possible the body shape of some of their semi-acoustics could end up on either a Gibson or an Epiphone. But the moment the neck had to be glued in, they knew exactly if it would be a Gibson or an Epi.
There's no way they shaped the headstock after the entire instrument was completed.