Sora and Riku have been best friends since very early in the childhood. They had a friendly rivalry and always sparred together with their wooden swords yet they still had a great admiration for each other. Sora and Riku both shared the desire to travel to other worlds and this especially became the case after Kairi arrived. They decided to build a raft together but were eventually sucked away from Destiny Islands by the darkness.
As soon as Sora and Riku ended up in separate worlds, they wanted to find each other. They eventually ran into each in Traverse Town. Sora was happy to see him but Donald said that Riku could not join them so he ran away. Maleficent tried to convince Riku that Sora did not care about him anymore and seeing Sora smile with new friends made Riku angry, sad and lonely, so in that moment of weakness, on his moment where he thought Sora didn't care about him, Maleficent could easily manipulate him.. Riku began to side with the heartless which angered Sora. They fought multiple times in Hollow Bastion and Riku temporarily stole Sora's Keyblade until he overpowered his heart and got it back.
Riku was upset that Sora's heart overpowered his and allowed Ansem to possess him. Sora eventually figured this out and demanded that he gave Riku his heart back. While the closing the Door to Darkness, they saw each other again. Riku helped him close the door and told him to take care of Kairi. Sora decided to continue to look for Riku. He went to Castle Oblivion and thought that he found him but it turned out that it was just a replica of Riku.
Sora was eventually put into a coma as a result of having his memories tampered with. Riku was determined to help him wake up and learned about his nobody Roxas. He fought Roxas in an attempt to capture him but could not win. Riku decided to give into the darkness in order to win which transformed his appearance into Xehanort's heartless and temporarily caused his memories to fade. He watched over Roxas until he returned himself to Sora.
Since Riku had given in to the darkness and changed his appearance, he did not want Sora to find him. While Sora was traveling, he dropped him some clues in order to help him move on. Sora and Riku eventually found each other in the World that Never Was and had a heart-warming reunion. They were eventually the last ones there and defeated Xemnas together. Sora and Riku were trapped in the Realm of Darkness together and had a heart-to-heart conversation. Riku admitted that he had always been jealous of Sora but the best thing about him was having him for a friend. The Door to Light eventually opened and they returned to the Destiny Islands together.
Sora and Riku spent some time at home until they received a letter from King Mickey. They were both called to take the Mark of Mastery exam by Master Yen Sid. Sora traveled through the Realm of Sleep while he was supposed to while Riku was actually traveling through Sora's dreams. They were determined to find each other and their actions still affected each other.
In the Symphony of Sorceress world, Mickey said that their Hearts were in tune, making them free to sing, while their "Heart Song" was Dearly Beloved.
It was eventually revealed that Xehanort intended to use Sora as a vessel and Riku had to save him. Afterward, Yen Sid named Riku aKkeyblade master and Sora was really excited for him.
In the Chain of Memories Light Novel, as well as the 358/2 Days novel, we get to see insight on Riku's take on the story, which hints that he may harbor more than friendship feelings for Sora.
"At that moment, I didn't even realize what was happening around me. I didn't even look. Sora, at the end of my outstretched hand, was so much more important than everything else."
"I wonder if I really will meet someone from my memories in this castle. I want to meet Sora, more than Kairi. More than anything, I want to see Sora. I want to see him-- and apologize."
Repliku was meant to show Riku's true emotions, what Riku couldn't express (How Sora abadonned them, etc.), so during the game and light novel, there are many scenes where he states that Sora always wormed his way into his heart, how he wants Sora to give him attention and such.
Even feeling content when Sora talked to him, and gave him "gentle words", knowing that he was just a replica, yet, he still got Sora to talk to him like that, and feeling jealous of the real Riku, for having Sora next to him.
In the Days novel, he thinks of Sora in order to regain his will to fight "Everything's for Sora", to the point he sacrifices his own body to the Darkness, if that meant Sora would wake up.
Sora doesn't fall too behind, as during the novels, his thoughts were on Riku only. Plus, this one line from the KH2 Novel, after their reunion.
"How hard did I look?
How hard did I miss him?
How much did I think, I want to talk to him again?
Riku is... here. I finally got to see him again, I missed him.
I missed him so, so much."
In an Interview with Nomura, he said that Passion was themed after Sora and Riku's reunion, the song, plus it's Eenglish counterpart, show a long of romantic hints.
"Following the previous game, Utada Hikaru also sang the theme song for this game, "Passion."Nomura: Many fans seem to have the image Kingdom Hearts = Utada Hikaru so I didn't think of changing that at all. This time I wrote a story explanation for her that's longer than the previous game's. We decided to have it played at the last scene, so for me the theme was "Sora and Riku's reunion." I wrote stuff like "An image of a reunion, a happy moment in a way" and "Like a Hikari 2" in there." - Via KHInsider
Taking a look at the translation/lyrics of both songs, you can see the romantic vibe song has, as well as Passion fitting Sora and Riku's childhood.
"I need more affection than you know."
- Probably hinting at Riku craving for Sora's attention, like KH1 Riku and Repliku demonstrated.
"I watch you fast asleep
All I fear means nothing...
In you and I, there's a new land
Angels in flight
My Sanctuary, my Sanctuary, yeah
Where fears and lies melt away...
My fears...my lies... Melt away..."
- Watch you fast asleep, probably hinting at the time in Days, where Riku waited for Sora to wake up, and that his fears meant nothing when saving Sora.
"When I remember far far back ''The future was bright for as far as I can see
Underneath the beautiful blue sky
We would keep sleeping'The person I loved a long time ago,
It appears a child is going to be born in the winter
Sometimes I want to have doubt
Over the settlement we had in the past."
Soriku is a very popular Kingdom Hearts ship and is possibly the most popular ship of them all. A lot of fans shipped this over SoKai from the start. However, this ship especially became very popular after Kingdom Hearts II. This was the first game where there was no ill feelings between the two and their relationship was emphasized more. Many fans feel that there is no way that the two are just friends.
Due to the later games and interviews (specially Dream Drop Distance), the fandom is speculating that Riku could be a possible love interest for Sora (Or at least, one-sided).