WBUR has been "Boston's NPR News Station" for a long time, I know it had been the slogan for a least a few years before I started working there in 1996. IIRC, it was in the late 1990's that WGBH started going by "Boston's NPR Arts & Culture Station", and prior to that they were "Boston's Classical and Jazz Outlet" (or something similar to that). The "NPR Arts & Culture Station" slogan persisted for many years, until perhaps this year? I don't know for sure, but they were still using it when I moved out of Boston in late 2007.
I'm not going to pretend to know everything about how to properly brand/market WGBH to their audience, but I can't deny that "Boston's NPR Station" feels like a weak attempt to capitalize on WBUR's success. I would've been more impressed by a switch to "Greater Boston Public Radio" or even "Massachusetts Public Radio" Yeah, obviously they neither reach, nor care, about anything west of I-495...if that far...but WGBH *does* have signal over most of eastern Massachusetts now, so I could forgive them the largess.
The "(location) Public Radio" concept usually works pretty well in the branding department for NPR outlets; there's a reason why VPR, NHPR, RIPR, and CPR all use it. (That'd be Vermont, New Hampshire, Rhode Island and Connecticut) Maine, too, although they go more by "Maine Public Broadcasting" because of their TV side. That's true even though CPR by no means covers all of Connecticut and has a lot of overlap with WSHU, much like WUNC (North Carolina Public Radio). Not to mention WAMC (North East Public Radio) and WSLU (North Country Public Radio), and, yes, WEOS (Finger Lakes Public Radio)...among many, many others.