Or indeed that 'ancestry groups' say less about which ethnic groups a person primarily descends from, and more about which ancestry groups are particularly prestigious. People who identify as Norwegian-American rarely have solely Norwegian ancestry, it's a statement of self-identity more than it is a statement of actual descent.
It doesn't have to be about prestige. For instance, Joe Biden, in weaving his own personal narrative, presents himself as an Irish Catholic in the vein of the Kennedys. But the majority of his family lineage is actually English. And generally, Americans have regarded having English ancestry as more "prestigious" than Irish ancestry.
But there is basically no such thing as "English-American" culture, nobody knows what that is, and it's not particularly interesting. Presenting himself as Irish allowed him to compare himself to popular political figures like the Kennedys, and present himself as someone who came from humble beginnings.