There are different models of how religion and society interact with each other. The American model is not universal, and Americans sometimes are confused about the relationship between religion and society in other cultures. Nevertheless, the American model is robust and seems to be capable of powerful assimilative feats.* In the early 19th century the Roman Catholic church was rapidly Americanizing in a manner we would recognize today, but the enormous influx of Irish and German Catholic immigrants at mid-century reversed this process. The Irish dominated hierarchy attempted to force the American political order to accept a form of official pillarization, but by and large this failed. Catholics did form a separate stream of civil society, but the persistence of a religious subculture seems to have been a function of the constant stream of new immigrants. Over a century later American Catholics, excepting a small “traditionalist” minority, have transformed themselves into another denomination.
A very similar process occurred with American Jews, though due to their small numbers they never faced-off against the Anglo-Protestant elite in the manner which the Catholics did. Orthodox Judaism, which most Jews around the world, secular or religious, would recognize as Judaism, is a minority faction in the United States. Rather, the more acculturated Reform and Conservative movements dominate. The Reform in particular has a long history of attempting rather consciously to transform itself into another Protestant denomination in form if not belief (though with the liberalization of mainline Protestantism there has been some convergence with Reform Jewish religious ideas).
The Japanese Americans who remain Buddhists (most of the community converted to Christian or are secular) adhere to the Buddhists Churches of America. And so on. Now the same with Hinduism, Old Faith Innovates in a New Land:
Ganesha is revered as the remover of obstacles, and his festival is considered an auspicious time to begin new endeavors, not least an experiment in adapting an old religion for a new land. And of the singers, most of whom grew up in India, none had ever heard of a Hindu choir before.
Choirs are virtually unheard of in temples in India because worshipers tend not to cohere into anything resembling an attentive congregation, said Vasudha Narayanan, a professor of religion and the director of the Center for the Study of Hindu Traditions at the University of Florida.
Some religious people get offended when I contend that it is the fate of all American religions to turn Protestant (non-Protestants that is). But unless you seal yourself off such as the Amish and Hasidic Jews have done to various extents, or replenish yourself with unassimilated immigrants, this is what simply happens. As someone not invested in any particular religious belief I generally think it best that the religions of the United States operate in a common cultural currency, the currency of confessional denominationalism. Even religions devoid of a creed such as Unitarian-Universalism wear their New England Congregationalist (ergo, Protestant Christian) origins on their sleeve.
Addendum: Though to be fair, even within the United States it seems that Greater New England and the South have developed in two very different trajectories when it comes to their interpretation of the appropriate exterior forms of Protestant worship and organization. It would be interesting to see if non-Protestants in these regions reflect these differences between Baptists and Congregationalists, for example.
* I would contend that the American model has been successfully planted in South Korea, much of Africa and parts of Latin America.