It actually makes more sense for Captain America to be for registration and Iron Man - given the personality established for Tony Stark - to be against it.
It’s always been that way in the comics too. When I read it I was pretty surprised, however it’s leftist drivel and I really didn’t enjoy that run very much.
Then again, Iron Man III, as much as I didn’t like that movie, could have created a genesis in Stark’s mind to believe registration of super powered people is a necessity.
It’s been established, through Captain America II, that Rogers will burn down organizations before he will trample the Constitution.
As for Stark, I can only guess that as a Genius, Billionaire, Playboy, Philanthropist, he's got a business angle in this, government contracts, and all that. But, yeah, it doesn't seem like him at all. He's more of a loner and didn't acknowledge Cap as the leader until they were in the middle of Manhattan with all Hell breaking loose.
Other way around.