The E-D is "Flagship of the Federation" in a couple of episodes and the movie ST:Generations. That's obviously not a military concept at all, because of the "Federation" rather than "Starfleet" bit there. Rather, the E-D apparently is the showpiece vessel of the Federation, being sent to impress the natives in various diplomatic contexts and whenever a bully needs bullying back; we see this happen often in TNG.
This best represents my view of the situation.
In contrast, Kirk's Enterprises never were called the flagship of anything. Neither were the E-B, the E-C or the E-E, or Archer's pre-Federation starship Enterprise.
While these ships were never spoken of as the "flagship", we didn't see very much of the B or the C, and I suspect that the Enterprise started to be considered the flagship during or immediately after Kirk's time. Hence the unusual continuation of the name and registry number, and the media hoopla during the launching of the B.
Also, in the context described above, I imagine that Archer's Enterprise would have been considered the flagship by default (whether or not the specific term was used), given that it was the first Warp 5 vessel and inaugurated a new era of human exploration.