Yeah, you can see this even from their titles:
The Manderlys are grateful to House Stark for taking them in when they were cast out of the Reach, and they’ve established themselves as a northern power, but I think there’s still a strong feeling within House Manderly as being a displaced southron House, rather than a born northern one. After all, it would be a difficult thing to abandon completely the faith and cultural identity that had been with the family for thousands of years before they first laid eyes on White Harbor. The Manderlys were too long at the center of reacher politics for that; there are too many centuries of Manderly history in the courts of the Gardener kings, too many old family heroes remembered for their deeds in the Reach, too many ancestors buried in graves by the river that still bears their name. They would still be reachlords today if not for the scheming of those no-good-very-bad Peakes, as they might think, and while they love the Starks for taking them in when no one else would, too much of the Manderly heart is probably still in the Reach. To give up the Seven and chivalric tradition might seem a deep betrayal of those old ancestral ties – a rejection of their identity.
You stand before Wyman Manderly, Lord of White Harbor and Warden of the White Knife, Shield of the Faith, Defender of the Dispossessed, Lord Marshal of the Mander, a Knight of the Order of the Green Hand,“
Yes, Lord of White Harbor and Warden of the White Knife are recognizing where the Manderlys are now, but look at the rest. Shield of the Faith tells you straight-up that the Manderlys are not going to convert to the Old Gods any time soon, Defender of the Dispossessed shows you how they are very much still mad about losing Dunstonbury, and the fact that they still refer to themselves as Lord Marshal of the Mander more than a thousand years after having to flee from the Reach and Knights of the Order of the Green Hand three hundred years after that entire order perished on the Field of Fire demonstrates how much the Reach remains in their minds.