Not all, but many religious people, especially conservative Christians, go a step further than that. They're not just proud of their beliefs identifying them, they take a sense of superiority from it. You don't have to spend much time around conservative Christians to know that they believe they are better than people who don't share their beliefs.
They'll be quick to tell you that their religion is the only real truth in the world, and many will even say that other belief systems are a "path to destruction."
Justice Samuel Alito just got recorded saying he agrees that the U.S. needs to return to "godliness," meaning (evidently) to return to a system of values and laws based on his personal religious beliefs.
He hasn't hesitated to do that in certain legal rulings, where he takes the moral superiority of his version of Christianity for granted.
It would be nice if we could have some sort of movement that values pluralism and a secular society — where anyone can choose religious belief, but nobody can impose it on anyone else.
Atheists and plenty of religious people could get behind that, though it would appeal mostly to atheists, I suspect.