I’m going to break this ask into parts to make my rambling more manageable. Let’s start with Sansa Stark, because I am always ready to talk about my baby goddess princess Sansa Stark. Also Catelyn, because Catelyn is great. Spoilers for the first three book (roughly equivalent to season 1-3; I don’t get to the season 4 or book 4 and 5 stuff)
People who don’t like Sansa or Catelyn will frequently say that they aren’t real Starks. Something in them just isn’t sufficiently Stark-y. The people saying this always mean it in a dumb misogynistic way (note: no one ever says this about 100% PURE TULLY ROBB STARK), but the thing is they’re not totally wrong. Before we get to Sansa, or even Cat, we have to talk about Ned.
Ned Stark was never raised to be the Stark in Winterfell. His elder brother Brandon was heir, and he was raised to play a completely differently role in life. Ned Stark was raised by Jon Arryn, and the brother he grew up loving was Robert Baratheon. When he was what – 16? – he married Catelyn Tully. Ned may have had the blood of the North in his veins, but his life was in the South. Catelyn talks about becoming accustomed to Winterfell and the ways of the North in her chapters, but Ned had to do the same. Catelyn becomes a Stark, but she and Ned both have a lot in them that’s more Tully and Arryn than Stark. Family, Duty, Honor – those are the actual values both Ned and Cat live by, and those are the values most of their children absorb. This is especially true of their eldest two children, the ones who look just like their mother and almost nothing like their father: Robb and Sansa.
This is not to say that Cat, Ned, Robb and Sansa don’t have any typical Stark qualities – they do! But are the elder Stark men the platonic ideals of Stark-hood that they’re often thought of as? Certainly not. (I’d actually argue that Cat absorbs more than either of those children and maybe even more than Ned). As Cat is shaped by the Starks, the Starks - including Ned - are shaped by Cat. “Family, Duty, Honor”, is a lot closer to “As High as Honor” than “Winter is Coming” is to either, and so Cat’s learned Tully values take root in Ned more easily than the Stark values of his dead father and brother.