I have experienced several somewhat extreme forms of multicultural Americans: Amish and Mennonites, Hasidic Jews, neighborhoods entirely made up of Latino immigrants or Korean immigrants and even, in some wards, neighborhoods which are Italian-dominant. Some want to assimilate, some do not. The Mennonites and the Hasidim definitely do not, and just want to be left alone.
In 1880, I suppose my WASP culture (and it is a culture) was still the core, dominant Americanism. Now, for better or worse, it's just one tribe of many.
Eugene Volokh discusses The American tradition of multiculturalism