^ It does when they're creating an "origin story," and then presenting things like this as either pre-established, or just assumed. Yes, the other Enterprises were seen as prestigeous assignments. However, the TOS Enterprise was first seen in the midst of its five-year mission, and established as a ship with a history under the celebrated Christopher Pike, not to mention an early historic mission to the edge of the galaxy under Kirk. The Enterprise-D, though without a history, had an extra century of legacy behind the name, and more importantly, was under the command of the famed Jean-Luc Picard.
In this origin movie, though, Enterprise is just a name. There's no rational reason for Uhura to fixate on an assignment to this ship over any other Constitution-class vessel (and certainly not for her to start fixating at age 5, when the ship was likely no more than a set of blueprints). Nor, frankly, is there a rational reason for her, as a cadet, to decide she's going to risk insubordination charges by complaining that she didn't get what she wanted. And Spock, being even more rational, should have told her to stow her emotionalism and report to the ship to which she'd been assigned.