The top gif is from a scene in Dream Drop Distance called “Memory of a Promise”, it is a short scene that involves Axel and Roxas sitting atop the Twilight Town clock tower eating ice cream, reminiscent of 358/2 days. The entire dialogue of the scene goes as follows.
Roxas: Hey Axel, do you remember?
Roxas: You promised to the both of us.
Axel: What is it?
Roxas: No matter how many times we ran away, your would always come to bring us back.
Roxas: Got it memorized?
Axel: A promise between friends.
The second gif is from 358/2 days and the scene it comes from is named “Axel and Xion Face Off”. This is one of the more powerful scenes in the game because it shows the effects that the actions of Roxas and Xion have on Axel. Personally, I see Axel as a mama bear and Roxas and Xion are his babies, throughout the entire game he jumps through hoops to defend the both of them and make sure they stay out of trouble. He bends over backwards to make things better for them, and how do they repay him? By running away from the Organization and leaving him behind while they “selfishly” search for answers to questions that Nobodies shouldn’t even be asking in the first place.
Axel is angry he says “You both think you can do whatever you want, well I’m sick of it. Go on, you just keep running. But I’ll always be there to bring you back!” In the heat of the moment Axel displayed his dedication, he did not want his friends to leave him, he did not want to be the one to have to destroy them, he wanted a simpler time to come back, he wanted it to be just like the days where the three of them could hang out without a care in the world again.
Think about Axel’s life in the organization before Roxas and Xion came along, his relationship was Saix was probably already deteriorating, so while they still had their mysterious plan their friendship was no longer as strong as it once was. He introduced Roxas to ice cream on their first outing together and said it was a habit for him, so Axel’s days consisted of sleep, missions, and ice cream atop the clock tower. Pretty bland if you ask me. But the this kid comes along and he’s just a zombie at first, so Axel, being the nice guy that he is, tries to get him to open up and warm up to him, nowhere in his job description was that written.
Think about if just about anyone besides Axel was assigned to “babysit” Roxas, they all would have been strictly business, teaching him the ropes then leaving, except for maybe Demyx who would con him into doing the mission and then taking him back to listen to him jam on his sitar. Roxas was very impressionable at first, so having anyone but Axel take care of him would have lead to him likely becoming a mindless worker drone, which would have been ideal for the Organization. But no, Axel treated this kid to ice cream and talked to him like a real person, he wanted to befriend him and it’s because of that kindness that Roxas developed his personality. Along comes Xion while Axel is out at Castle Oblivion and Roxas passes on what Axel taught him to Xion, because that was all he knew, the two become close and when Axel comes back she asks if they can be friends too. He agrees and this happy little family is born.
To me what separates the Days trio from the other two is that in Days, the three of them rely on each other to get through life. They work hard so that at the end of the day they can reunite and have ice cream and laugh and just be a family. Which is why it hurts Axel so much for those two to leave him. He doesn’t want to go back to the empty days.
But listen to what Roxas says, you made us a promise. You made us a promise. While it’s not clear if Axel remembers Xion clearly or not after her death, the imprint of her is still there. Neither in Japanese nor in English did Roxas say, you made me a promise. Even after his rebirth, Axel, now Lea sets out to fulfil his promise. And while Roxas and Xion may not physically exist any more they are still alive inside of Sora, so it falls to him to protect his friends inside of Sora. And he does just that.
While I don’t know everything about KH:3D just yet I do know that while the story focuses on Sora and Riku, it’s Lea who steals the game.