White Americans all have different ancestries.

That's true, but it's not all that relevant to Americans. To the extent that groups of white Americans regard themselves as different from other white Americans, its a geographic and cultural thing - "The South" vs "The Midwest" vs "The Northeast", not "Scots-Irish" vs "Scandinavians" vs "Anglos" even if those are largely two ways of looking at the same thing.

Right now, I think most white Americans see themselves as basically the same as other white Americans because the gap between Scots-Irish Southerners and Anglo Northeasterners pales in comparison to the gap between whites and the nonwhite races in America.