The United Federation of Planets and Starfleet are intertwined, but the two terms aren't necessarily synonymous. In the Star Trek timeline, both organizations were founded in the early 22nd century, but Starfleet predates the Federation by a few decades. Starfleet began as a human-founded body dedicated to space exploration, whereas the Federation had four founding species: humans, Vulcans, Tellarites, and Andorians. Starfleet went on to enlist non-human officers too and was eventually incorporated into the Federation's infrastructure. Both bodies continued to grow and change before, during, and after Picard's spell as captain of the Enterprise.
Why Captain Picard Introduced His Enterprise As A "Federation" Starship, Not Starfleet In Star Trek: TNG
The USS Enterprise-D is both a Starfleet AND a Federation ship
The USS Enterprise-D was far more than a standard Starfleet vessel. It was the Federation's flagship, which meant it often represented the interests of both organizations. It wouldn't have been inaccurate for Stewart's character to introduce himself as the captain of a Starfleet ship, but it wouldn't have painted the entire picture either. Plus, the Federation's flagship also boasted personnel without direct ties to Starfleet, but their scientific research and other endeavors were carried out in the name of the Federation. It was a collaborative crew, so it would've been disingenuous for Picard to introduce his ship as strictly Starfleet.
Of course, Picard isn't the only Star Trek character to identify his Starfleet ship as flying under the Federation banner. In the 23rd century, Spock (Leonard Nimoy) can be heard naming Kirk's ship as a Federation vessel when hailing the Tholians in The Original Series season 3, episode 9, "The Tholian Web." Captain Janeway (Kate Mulgrew) also does it in the Star Trek: Voyager premiere when introducing herself to Neelix. So, it would appear to be a procedure that predates Picard, and it's also not exclusive to him. That said, it's arguably more important that Picard is consistent with it.
The Difference Between Kirk & Picard's Enterprise In Star Trek
Both vessels had very different journeys to becoming the Federation's flagship
From a canonical perspective, Picard's Enterprise was the Federation's flagship from Star Trek: The Next Generation's inception. However, Captain James T. Kirk (William Shatner) was only retroactively given the honor of commanding the Federation's flagship. Kirk's Enterprise was retconned into such a position over a series of franchise installments, starting with JJ Abrams' first Kelvin Timeline movie, 2009's Star Trek. With the iconic role inherited by Chris Pine, Abrams made the alternate version of Kirk's Enterprise the Federation's flagship.
Star Trek: Discovery and Star Trek: Strange New Worlds came along, and it was officially confirmed that Captain Pike's (Anson Mount) Enterprise, which would go on to become Kirk's, had actually been the Federation's flagship all along.
Abrams' alterations arguably became canon in Star Trek's Prime Universe too, but it was difficult to prove. Eventually, Star Trek: Discovery and Star Trek: Strange New Worlds came along, and it was officially confirmed that Captain Pike's (Anson Mount) Enterprise, which would go on to become Kirk's, had actually been the Federation's flagship all along. This retcon also explains why Kirk's Enterprise was kept out of the Klingon War. Picard's opening hail in Star Trek: The Next Generation didn't hold quite so much meaning at the time, but it certainly does now.
