"American" is not an anthropological category, though.

Consider how odd "ethnically American" sounds to the year. Compare that with "ethically Chinese", "ethnically Indian", "ethnically Korean".

"American" is a geopolitical concept defined in terms of a political entity, the United States (just like "Israeli" is a geopolitical concept defined in terms of the country Israel). This is just a consequence of the fact that America is not a nation state (In this respect compare e.g. the -stan countries of Central Asia, each of which is named after a dominant ethnicity. "Tajik" is not defined in terms of "Tajikistan", rather the reverse is true).

Native Americans are ethnically American.

Is "ethnically American" even a single homogeneous thing though?

How many languages are spoken, how much territory used to be part of the New Spainish lands for how many hundred years, how many diaspora's fed into modern America?

"Native American" itself is a broad spectrum, many past language and cutural groups, many degrees of living on and off reservation lands, many intertwinings with former slaves, ranchers, et al.

While I follow the argument, wouldn't by necessity, an ethnic Americans begin to exist the moment the next generation is born ?

tldr: Anthropological quality has to start somewhere; not completely unlike myths.

I wouldn't say ethnic-group formation happens as soon as the next generation (after the founding generation), but I agree with the larger point. I wouldn't be surprised if in a couple hundred years no one would bat an eye at talk of "ethnically American" or "Norwegian of American descent". My understanding is that that's more or less what happened with the French.