Riku was Sora's. I mean, technically he was even owned by Sora's family. But...his heart belonged to Sora, and that was what really mattered.
Sora clasped Corin's hand and gave it a squeeze before letting go. "I am serious, we will be family so..." It felt a little awkward reaching out to him like this, and Sora almost felt a little vulnerable. "So please trust me." What he really meant was please accept me. It was important to him.
"Come! Let us return to the festivities," Sora suggested, sweeping his arm to indicate that Corin should go first. He planned to have Riku spend plenty of time with Corin, and he did really want Corin to see how WELL Riku filled his role.