The X-Men films typically follow mainstream knowledge of the comics. Wolverine wasn't known to have bone claws by the general audience, so they didn't mention it. The Bone Claws reveal was only done in 1993, recent compared to Wolverines entire history. He's typically depicted as thinking that he gained claws from the Weapon X program.
Wolverine has always had memory issues. He normally doesn't know who he really is, or his childhood. Readers didn't find out he was James Howlet until the Wolverine Origin comic in 2002, which was then used as the basis for the Movie. That revealed that Wolverine always had bone claws as far back as childhood, and Sabertooth was his blood brother.
X2 was released in 2003, so incorporating anything from 2002 into the script would have been unlikely.
Notice that Wolverine is confused in Days of Future Past, when he wakes up the 60s and has bone claws instead of metal. His memory was erased at the end of Origin by an Adamantium bullet to the brain, courtesy of Striker. Xavier is shown to be working to help Wolverine regain his memory through out all the movies.
Assuming all of this is in continuity, Stryker has been shown to be quite smart, cunning and willing to use people, and having known Wolverine suffered memory loss, his wording can be assumed to be an intentional omission. He admits what Wolverine already knows, and doesn't volunteer what he doesn't.