First off, the Enterprise is not a naval vessel. It is a Starfleet vessel. Regardless of some of the similarities, Starfleet is not the modern military. They could certainly use terms differently in the 24th century.
Yes, they could, but this is a case where I wish they wouldn't. I'm not saying it's impossible, since clearly canon shows that it is used that way. I'm saying it was a bad idea on the part of those writers who chose to incorporate the concept. The whole "Enterprise
as flagship" notion is one of those things that started in fandom and then got injected into later productions, and I don't think it should've been, partly because it's too fannish and partly because it's an awkward use of the word "flagship."