But all those places are called Berlin, New Hampshire; Manchester, New Hampshire; Dublin, New Hampshire, etc.... The analogy of different planets having the same name falls apart if you consider the ENTIRE name of the planet. Generally, the nomenclature is "Star Name"+"Planet Number" (Ceti Alpha V, Sol III, etc...). Presumably "Delta Vega" should really have been "Vega Delta" (fourth planet of Vega), which pretty much restricts how many planets can have that name. Unless you want to argue that "Delta Vega" was a famous figure in Federation history, and it really doesn't refer to any particular star system.
Of course, whoever wrote WNMHGB probably had no forethought about what they were writing, anyway ("James R Kirk"), so that's excusable. And I agree the name was simply co-opted for reference-sake in ST09. But really, they should just have picked a better name -- perhaps one of the moons of Vulcan, considering it was in visible range of the planet (but Vulcan had no moon -- until TMP showed it to have several -- so...). Calling it Delta Vega really makes as much sense as calling it Mars.