No, the Keyblade in the first game is definitely Riku's. The Key is first summoned by Riku, but he gets swept away to Hollow Bastion so it automatically goes to Sora.
When they meet up in Traverse Town, Riku is able to hold onto the Keyblade without Sora being able to take it back (like he does previously with Leon) because it belongs to Riku, but he gives it back.
Later in Hollow Bastion, he's able to summon it off Sora because it belongs to him, but it eventually chooses Sora over him because he has the stronger heart.
I'm pretty sure the first game explains this, but if you haven't played it in awhile it'd probably be an easy detail forget.