I think Arya often gets labeled the “Stark” because she’s so often physically compared to Ned and Lyanna, but I absolutely think Arya is fully Catelyn’s daughter. I’d even go so far as to say Arya is more like Catelyn than Sansa is but I don’t think most people would recognize it, including Catelyn and Arya.
There are references through the book from Ned’s POV that he wasn’t quite like his siblings and the wild wolf blood, and it’s implied Arya has that. But we also know Edmure and Lysa are impulsive with tempers, the Blackfish is awesomely stubborn, Hoster obviously had conflicts with his brother, and Catelyn is awesome pretty much in every respect. Like Catelyn, Arya is intelligent, resourceful, and comfortable holding her own among men; she lacks Catelyn’s political acumen and courtesies, but that’s because she’s a child and wasn’t raised in the South where courtly manners seem to be stressed much more. Though Arya seems to identify more with her father and the Northern customs, she repeatedly stresses that she prays to the Old Gods and the New. As shown in ASOS, Arya misses her mother and longs to return to her and save her, nearly getting herself killed in the process. Catelyn, at great political cost to Robb and the North, let Jaime go so her daughters could be returned to her. But I think what Catelyn and Arya share the most is a frustration at the limitations placed upon them by their sex.
We see this most clearly when Catelyn calls Robb to the carpet for not trading Jaime. ”Girls aren’t important enough.” Catelyn knows her daughters have essentially no value in Westerosi society beyond who they can be married to and what that will bring their house; though her marriage turned out well, Catelyn was promised to Brandon Stark when she barely a teenager and then, when he died, just got passed to his brother like a horse. She saw Lysa married off to a man old enough to be their grandfather. Sansa was promised to Joff, and the North is now in open rebellion against the Iron Throne, so that betrothal gives Sansa no protection and Arya, even less. By Robb refusing to go for his sisters, even though it is technically the smart decision politically, it is a reminder to Catelyn of how little women are valued. And as we see when she starts giving Robb advice he does not want to take, she says kings do not want mothers. Even though Catelyn is arguably the best politician among Robb’s allies, her voice matters less because she is “just” a woman.
Compare this to Arya, whose first order from Yoren is to disguise her sex so she isn’t raped on the road. While Sansa has been sexually intimidated and assaulted, it is Arya who has been told to constantly be on alert for the possibility of it because she is among “criminals.” (And the sexual assault themes in this series is an entirely different kind of meta.) The only women Arya sees are ones who are abused (Pia), sex workers (the women at the Peach), and mothers (the Lady of Acorn Hall, Catelyn), all of whom fit scripted roles in Arya’s admittedly limited experience. For Arya, she sees Catelyn as the woman who tends to her children, is married to her father, helps run Winterfell; she doesn’t see Catelyn as possessing any real power, and the only woman she’s ever seen with power (Cersei) wronged her and her family so egregiously, Arya wants to murder her. There’s a lot of meta about Arya hating women and the fluidity of Arya’s gender presentation, but I don’t think Arya hates women or even that she wants to be a man; I think Arya resents the limitations put on her, doesn’t understand the incredibly limited amount of options available to women so when they don’t fight back, she gets doubly frustrated, and knows she is safer when no one knows she’s a girl because she cannot be victimized.
The moment the Brotherhood finds Arya, what’s the first thing they do? They force her back into a gown and she’s back to being Lady Arya. The freedom and ability to protect herself is taken away; even wrestling with Gendry in the forge gets Gendry chastised because you don’t behave that way with a lady, never mind what Arya wants. In ASOS, we see Arya and Catelyn both stripped of the agency they’ve acquired in ACOK, Arya by the Brotherhood and Catelyn by Robb and his men.
Even the Red Wedding shows us how similar they are. When Arya knows her family is in trouble, she runs towards the Twins even though it would mean certain death. When Catelyn sees Robb and his men being killed around her, she tries to intervene and murders Jinglebell even though it means certain death. Family, duty, honor. Even in Braavos when Arya is technically free of any Westerosi expectation placed upon her, she still kills Daeron of the Night’s Watch because he broke his vow, he soiled his honor and it’s the duty of the Warden of the North to execute deserters.
And I also think it’s worth mentioning that it’s Nymeria who pulls Catelyn’s body from the Trident, who allows her to be revived by Lord Beric. We know Arya’s a warg, and we know the “wolf dreams” are the Starks in wolf bodies; even if Arya doesn’t understand it, she pulled her mother from the river and delivered her to the people who wanted to protect Arya and take her to her family.
Lady Stoneheart and Faceless!Man!Arya aren’t that different of characters. Arya is No One; she is an echo of whoever wore her face last, she is whoever the priests tell her to be in order to give the gift. Lady Stoneheart is not Catelyn Stark; “she don’t speak but she remembers.” She has a list just like Arya: Lannisters, Boltons, Freys. The manifestations are different but the goal is the same: revenge. When you think of Ned and Sansa, they didn’t/don’t want revenge; they want to go home, to escape a world that doesn’t make sense. Arya and Catelyn/Stoneheart know there is no home to return to so they’re going to take everyone else down with them.
And that’s why the lack of focus on Arya and Catelyn’s relationship is criminal.