...Can we even be sure that the title "Federation Flagship" is unique, definite article or no?
If Starfleet starships exist in ten's of thousands, then it would make sense that there would be many "The Flagship of the Federation." Divided (or not) among the Federation's various lanes of expansion. They wouldn't all have to be Galaxy's, but likely would still be a starship that was capable of being dressed to impress.
I think there are still ships that serve as flagships in the traditional sense, particularly in taskforce operations, and such vessels are probably referred to as "the Federation flagship" by an objective party, but I also think the Enterprise
Federation flagship--the definite article--and did hold a special distinction above all other vessels in the Starfleet.