A different time, and a oft misinterprited phrase deliberatly inserted to distinguish us from European nations with an official religion. And they were correct, we don't have an official religion, however I don't think the circumstances of Obama's speech are remotely similar. What he meant I haven't a clue, it probably popped up on the teleprompter and he read it.
The country was founded by Christians and the universalist principles—all created by equally the Creator—stem from a Christian ethos. The fact that others—Hindus, Jews, Bahais etc.—came and thrived here is in fact due to this universalist ethos. Obama is somewhere between a Muslim and a nonbeliever. He is obviously not a Christian, but the country is. He can’t change history.
He’s trying to be nice with the Muslims. It’s the same circumstances.
But as you mentioned, what meaning he personally had could be completely different from what was meant in the treaty, which was “We are not your enemy because of any religious nature of this country.” IOW, we’re not going to try to wipe out Muslims out of some religious fervor. It was a sensible statement in the treaty.
But with Obama you never know.