I think originally the “Anglo Saxon” bit meant that they were derived from British immigrants to the US, not those other groups.
I’ll venture a guess though and say that in most of the country where there wasn’t mass Irish or Eastern European immigration in the 19th century, all those heritages are lumped into the generic “white”, with all rights and privileges pertaining thereto. And that’s what this guy means by “Anglo Saxon”, I suspect.
Plus, there’s been a lot of intermarriage among those groups outside of certain ethnic-ish enclaves on the East Coast. I mean, I can use myself as an example. Probably the largest single ethnic group in my background is English, but there’s an almost equally large chunk of German background- both Texas German and Pennsylvania Dutch. There’s also some relatively recent French and Scotch-Irish along with a bit of Native American.
I’m as white as can be, but not really of English or German descent; at least not enough to really feel like I can claim either anyway.