I guess that about 90% of Americans actually born in the States have mixed origins.
Myself, as a European, have various origins. But it all depends what people consider as "mixed". For example could someone who is half British, half Australian (of British descent) be considered as "mixed" ? Nationalities are different, but the ethny and language are the same.
In Europe it's very difficult to separate "ethnic groups" because most countries have invaded each others or have different borders throughout history. Modern nationalities don't reflect at all "ethnies". It is estimated that about 50% of Northern French people have Germanic blood. Germanic tribes who invaded the Roman Empire left descendants in all Southern Europe and the Maghreb, and so did the Vikings a few centuries later.
My Genealogy, though it doesn't go as far back as the Vikings or Roman times (which is anyway impossible), shows that I have French, German, Belgian, Dutch, English and possibly Italian ancestors.
But of course my g-tree isn't complete. If I wanted to know where
all my ancestors (i.e. all my DNA) came from 300 years ago, that would be about 12 generations (25 years per generation in average), which in other words is 4096 people of this generation only. So I am not exactly sure what percentage of these 4096 people exactly came from where, but that would be interesting to know.
In addition to this multiplicity of origins, I am proud to say that I have the nationality of
each of my ancestors' country.... Don't believe me, huh ? Well since 1992, the European nationality has been created for the EU in addition of individual countries. That's suits me very well, and I hope that only the European nationality remain in the future (probably not before all European countries have joined the EU, adopted the Euro and reinforced the EU government, but that could happen quicker than one expect).
I feel the cultural heritage of all the European countries I cited, as I have lived in or learned the language of each country.
Even if they have been ethnically mixed for ages, Westerners have a very rich and diverse cultural heritage, as each linguistic group is different culture. No matter whether they are Europeans, Americans (all continent), Australian or South African, most Westerners have mixed European origins. That is probaly what makes it so difficult for Japanese to distinguish between them.
For me the "half-this", "half-that" issue is not relevant, except if only the parents and the knowledge of their language and culture are to be taken into account. "Race" doesn't mean anything. Some Japanese are mistaken for Chinese or Koreans in their own country, even when they are pure Japanese. I think that the language one speaks is reflected on one's face. I can pretty easily say who is British Indian and who is Indian from India, just from the facial expression. That's also how I tell European between them. Some French people, whatever they look like, have an expression that says "I am French" on their forehead.

It's usually possible to distinguish most (not all) Scandinavians between them, even though they are all "pure Germanic"and have the same origin and a similar language. With some practice it's possible to tell Koreans, Chinese and Japanese apart. But if they were born and raised in a Western country, that's much more difficult.