I'm lord of Moat Cailin and it still shows the Stark banner.
(https://i.gyazo.com/ab02d22aaca86e31b1c6c5e0d588975f.jpg)
I think this banner isn't a changeable banner and it will be stuck like that, there are a few which do the same thing. Such as the one from the old valryia quest always shows the Targ banner.