America's strength has never been its diversity, but its ability to overcome diversity through assimilation. "E Pluribus Unum" (Out of Many, One) refers to the thirteen colonies, but could as easily describe our melting pot.
It is no insult to other cultures to say that America has its own. We say we are a multicultural nation because we worry it may offend to say otherwise, but also because it appears true from our surface diversity (skin color, food, clothes music). Yet American diversity is a mile wide and an inch deep. Beneath the surface most Americans share a sense of nationhood and fundamental values (even if Reds and Blues accuse each other of betraying those values). That sounds vague because, like obscenity, American culture is easy to recognize but difficult to define. Yet its truth becomes apparent to any American traveling abroad, many of whom say they've never felt so American as when visiting their ancestral homelands.
Surface diversity is enriching, but deep diversity can be dangerously divisive. Despite their more homogeneous surfaces, diversity runs so deep in Northern Ireland, Rwanda, the Mideast, and the tellingly "former" Yugoslavia that people have murdered one another to assure the dominance of their religious or ethnic group.
America too has suffered deep diversity, Jim Crow being only one recent example. Yet like the Borg, American culture continues to assimilate everything so it belongs to everyone. Chinese take-out and Italian pizza are not evidence of our multiculturalism, but things we've all come to know and share in. We speak a common language, we increasingly vote and marry outside our ethnicities, and we have at least a passing familiarity with most elements in our common culture. For example, I've only seen a handful of Star Trek episodes beyond the original series, and none featuring the Borg. Yet American culture is so pervasive, I know enough of the Borg to use them in an analogy.
Another reason Americans confuse themselves for a multicultural nation is that identity politics conflate race and culture. Shown a multiethnic group photo, many will thoughtlessly exclaim, "Oh, how multicultural!" But unless culture is genetically transmitted, an ethnically Chinese girl raised in Germany is culturally German, just as an Italian boy raised in China is culturally Chinese. Likewise, families raised in America are culturally American. Yet by confusing race and culture, Americans are dissuaded from promoting their own culture lest they appear exclusionary by celebrating something they've been convinced immigrants are genetically incapable of sharing in. (No one puts it like that, but those are the implied undercurrents of identity politics.)