In both Jon and Arya’s first chapters in AGOT, George does a wonderful job of showcasing their similarities, specifically their personalities and how they respond to adverse situations where their pride is wounded by others.
We see this in Jon I when Benjen tries to tell Jon he doesn’t think it would be a good idea for Jon to join the Night’s Watch:
Benjen Stark stood up. “More’s the pity.” He put a hand on Jon’s shoulder. “Come back to me after you’ve fathered a few bastards of your own, and we’ll see how you feel.”
Jon trembled. “I will never father a bastard,” he said carefully. “Never!” He spat it out like venom.
Suddenly he realized that the table had fallen silent, and they were all looking at him. He felt the tears begin to well behind his eyes. He pushed himself to his feet.
“I must be excused,” he said with the last of his dignity. He whirled and bolted before they could see him cry. He must have drunk more wine than he had realized. His feet got tangled under him as he tried to leave, and he lurched sideways into a serving girl and sent a flagon of spiced wine crashing to the floor. Laughter boomed all around him, and Jon felt hot tears on his cheeks. Someone tried to steady him. He wrenched free of their grip and ran, half-blind, for the door. Ghost followed close at his heels, out into the night. […] The sounds of music and song spilled through the open windows behind him. They were the last things Jon wanted to hear. He wiped away his tears on the sleeve of his shirt, furious that he had let them fall, and turned to go.
(Jon I, A Game of Thrones)
Jon’s first reaction to Benjen is anger: being a bastard himself and growing up with that adversity, the idea of fathering a bastard of his own is the worst thing in the world to Jon. Drunk as he is, Jon is unable to hold his tongue, causing unwanted attention to fall on him by the rest of the group.
Once Jon sees this, he begins to cry and tries to excuse himself to preserve his pride. This doesn’t work out for Jon, however, and he only embarrasses himself further, and hates himself for it.
The next day, in Arya’s first chapter, she goes through a very similar experience while doing needlework with the other ladies:
“Arya, why aren’t you at work?” the septa asked. She rose to her feet, starched skirts rustling as she started across the room. “Let me see your stitches.”
Arya wanted to scream. It was just like Sansa to go and attract the septa’s attention. “Here,” she said, surrendering up her work.
The septa examined the fabric. “Arya, Arya, Arya,” she said. “This will not do. This will not do at all.”
Everyone was looking at her. It was too much. Sansa was too well bred to smile at her sister’s disgrace, but Jeyne was smirking on her behalf. Even Princess Myrcella looked sorry for her. Arya felt tears filling her eyes. She pushed herself out of her chair and bolted for the door.
Septa Mordane called after her. “Arya, come back here! Don’t you take another step! Your lady mother will hear of this. In front of our royal princess too! You’ll shame us all!”
Arya stopped at the door and turned back, biting her lip. The tears were running down her cheeks now. She managed a stiff little bow to Myrcella. “By your leave, my lady.”
Myrcella blinked at her and looked to her ladies for guidance. But if she was uncertain, Septa Mordane was not. “Just where do you think you are going, Arya?” the septa demanded.
Arya glared at her. “I have to go shoe a horse,” she said sweetly, taking a brief satisfaction in the shock on the septa’s face. Then she whirled and made her exit, running down the steps as fast as her feet would take her.
(Arya I, A Game of Thrones)
Like Jon, Arya is given unwanted attention by Septa Mordane and is embarrassed by the state of her needlework. Arya’s first reaction is also anger, “Arya wanted to scream,” but being a highborn lady rather than a bastard (not to mention being 9 years old and sober), Arya is able to guard her tongue (for the time being) when Jon was not.
When she is chastised by the septa, Arya’s response is identical to Jon’s. She hates that everyone is looking at her, that Jeyne is smirking at her, and begins to cry. Also like Jon, Arya can’t bear for anyone to see her tears, so she tries to flee. Only when Septa Mordane calls after he does Arya stop and extend her courtesies to Princess Myrcella, just as Jon excused himself at the feast to maintain some of his dignity.
As it so happens, after Arya flees the scene, she finds Jon watching their brothers training in the yard with Prince Joffrey, and the pair of outcast siblings comment together about life is not fair.