As Timo said, the TOS ship was never refered to as the flagship of anything. It was obviously the series'
flagship, given it was the one whose adventures we were chronicling. But in-universe it was always treated as though the Enterprise was just one of many ships, and that they all
were of equal stature within the fleet.
The Enterprise as a flagship didn't really begin until TNG (and as others have pointed out already, even then the term 'flagship' was used in a very loosely defined way).