When I read the title of this question, my first and foremost thought was: Does it matter? In the grand scheme of things, we shall see what we shall see: Bran is doing his thing, Arya her thing, Sansa her thing, and Jon his thing. What Rickon does, we do not know (yet).
My second thought was, yes, of course it did matter way back when, when I first read the series.
I've had a while to think about this, so perhaps I can review my thoughts.
In A Game of Thrones, the very first book, the very first chapter, the very first few pages, Jon Snow says to Lord Eddard Stark, regarding the direwolf pups found in the snow:
"You have five trueborn children," Jon said. "Three sons, two daughters. The direwolf is the sigil of your House. Your children were meant to have these pups, my lord."
Then we find out that there is another pup, one that is all white.
"An albino," Theon Greyjoy said with wry amusement. "This one will
die even faster than the others."
Jon Snow gave his father's ward a long, chilling look. "I think not,
Greyjoy," he said. "This one belongs to me."
It is clear from the very start of the book that this is supposed to mean something. I believe the first person to know how special this bond is is Bran, although I cannot safely say what order this happened. But I do remember the moment when Bran woke up after his long coma dreams, scaring a serving woman to death:
When his brother Robb burst into the room, breathless from his dash up the tower steps, the direwolf was licking Bran's face. Bran looked up calmly. "His name is Summer," he said.
We get countless examples of Bran dreaming "wolf dreams", which makes it redundant to elaborate on Bran/Summer. We also know that Jon has some intense moments where he and Ghost are connected in some otherworldly way, such as when he sees the huge wildling camp at the Milkwater through Ghost's eyes.
We also know that Arya has vivid dreams of Nymeria, such as when Nymeria swims out into the river and drags Catelyn Stark's body to the shore, and the next morning, Arya knows her mother is dead.
We also see Arya in Braavos, seeing through the eyes of a cat when the kindly old man is playing tricks on her, and know that she still is having wolf dreams.
Sansa is the only one who never has any wolf dreams, or visions through her direwolf Lady. We do know that Lady, like all the other direwolves, reflected her owners personality in her own: She was well-behaved and ladylike. She was killed rather early on, through Sansa's own cowardice, so we do not know much of that. We do know, however, that Sansa tamed another large and dangerous canine: Sandor "The Hound" Clegane. Read the chapters, I am sure you will see what I am talking about.
Then there is Robb. The forlorn lord, who lost the north. We do not know what goes on in Robb's mind. But we do know that the extra sensory perception that was evident in all the wolves: The ability to sense evil or betrayal in people, the ability to sense danger, or grief, was also present in Grey Wind. But Robb pushed aside his intuition and his trust in Grey Wind, when the untrue rumour of his brothers' demise reached him.
[Catelyn] "Any man Grey Wind mislikes is a man I do not want close to you.
These wolves are more than wolves, Robb. You must know that. I think
perhaps the gods sent them to us. Your father's gods, the old gods of
the north. Five wolf pups, Robb, five for five Stark children."
"Six," said Robb. "There was a wolf for Jon as well. I found them,
remember? I know how many there were and where they came from. I used
to think the same as you, that the wolves were our guardians, our
protectors, until . . . "
"Until?" she prompted.
Robb's mouth tightened. " . . . Until they told me that Theon had
murdered Bran and Rickon. Small good their wolves did them. I am no
longer a boy, Mother. I'm a king, and I can protect myself." He
sighed. "I will find some duty for Ser Rolph, some pretext to send him
away. Not because of his smell, but to ease your mind. You have
suffered enough."
Robb chooses not to believe. He is a King, and he lacks the confidence to believe in supernatural things. Despite Grey Wind being almost a part of him at this time, fighting beside him in battles, he allows the Westerlings and the Freys to get between him and his wolf. When he should have trusted his wolf, he does not, and suffers for it. An event that is also shadowed in the events surrounding Jon in ADWD.
Last but not least is Rickon and Shaggydog. It is clear to me that Shaggydog is very much in touch with Rickon's feelings. Rickon is the "wild wolf" in the family, and spending all this time with Osha and the wildlings (and unicorns!), the next time we see him, it is sure to be a lively event. The most remarkable event in this context is, I think, in Bran's POV, when Bran convinces Maester Luwin to take him down into the crypts to see if Lord Eddard is, in fact, down there. And instead they find Rickon and Shaggydog, and it turns out Rickon had the same dreams as Bran.
"Shaggy," a small voice called. When Bran looked up, his little
brother was standing in the mouth of Father's tomb. With one final
snap at Summer's face, Shaggydog broke off and bounded to Rickon's
side. "You let my father be," Rickon warned Luwin. "You let him be."
"Rickon," Bran said softly. "Father's not here."
"Yes he is. I saw him." Tears glistened on Rickon's face. "I saw him
last night."
"In your dream . . . ?"
Rickon nodded. "You leave him. You leave him be. He's coming home now,
like he promised. He's coming home."
So I would say, yes, all of the Stark children have some of the "wolf blood" in them. Bran the most, and Sansa the least. But Jon and Arya have strong experiences as well, and while we do not know the inner thoughts of Robb or Rickon, they do seem to share the same kind of bond with their wolves.
