At the end of the second game, Riku admits that he's always been jealous of Sora, and then...
Riku: But there is one advantage to being me. Something you could never imitate.
Sora: What's that?
Riku: Having you as a friend.
Sora: Then I've got something you could never imitate too.
The end of KH2. Sora: "We're... we're back." Kairi: "You're home." Also, when Sora gets tackled in a hug by Donald and Goofy and Riku catches Mickey and spins him around.
"I'm sure Sora will find the answer. Because he's me." Though it at first sounds odd, and maybe even egotistical, I can't help but smile every time I think of this moment from KH2: Final Mix. Through most of the game (especially with scene additions), Roxas makes it clear that he wants his own identity, that he doesn't want to just be Sora's Nobody. He lashes out at anyone who calls him Sora, and continuously says his heart belongs to him. But now he's accepted the fact that Sora is a part of who he is, but not all he has to be, and that they can learn to coexist without having to give up anything.
Pluto trying to comfort Kairi in the prison cell of the Castle That Never Was in KH2.
Mickey and Donald's reunion with Goofy after his Disney Death.
Sora and Kairi hugging in Kingdom Hearts II.
Tron's goodbye after defeating the MCP.
It's even more Heartwarming In Hindsight. Kingdom Hearts 3D [Dream Drop Distance] reveals that programs are somewhat like Nobodies, as they don't have hearts either. The same game confirms that Nobodies can grow their own hearts, given time, and it's quite possible that the same could go with programs. Tron is a program, and thus does not have a heart at first: his friendship with Sora allowed Tron to grow one of his own!
A little tiny one: After Mickey reaches the unconscious DiZ (who, of course, is actually Ansem the Wise) in The World That Never Was, the latter starts to stand up. DiZ stumbles for a moment, then glances at Mickey, who he's using to support himself. He pauses for a moment before standing, which this troper interpreted as giving his old friend a hug.
A rather nice little moment occurs while Santa and Sally are talking in Kingdom Hearts II. While chatting Maleficent and Bogie attack them in an attempt to capture Santa. Despite not knowing Sally for more than a few minutes, Santa valiantly puts himself in front of her and their attackers. Pure of heart? That's Santa Claus alright.
Another Kingdom Hearts II example. If (by some incredible chance) you lose to Setzer in the Roxas prologue, Roxas returns to his friends looking disappointed. The four are silent for a moment. Then Pence starts clapping, followed by Hayner and Olette. Seeing Roxas' friends support him like that just makes this troper feel happy.
In the second game, when Donald and Goofy finally get to reunite with Mickey at Hollow Bastion. They'd seen him before at Twilight Town, but this time they actually get to run up and hug him.
In the manga adaptation for Kingdom Hearts II, we get the scene of Sora allowing Demyx to use his Olympus Stone, despite the objections of his friends.
Another manga example is when Sora manages to snap Donald and Goofy out of a Heroic BSOD when the trio find themselves trapped inside a computer after being sentenced to death by it's Master Control Program, and their only means of escape is accidently destroyed.
Sora: Don't give up just yet! Have you guys already forgotten?! We made a promise to meet up with the King again. There's no way we're gonna be stuck here forever!