One set of great grand parents were Irish. Another set were English but of French descent. Does that make me Irish-English, French-English or, as I prefer, just English (or British, or European (for now) as I choose)
As my father used to say: "What it makes you is American. We're all of the Heinz 57 breed."*
*I and many others of my generation understand the reference to "Heinz 57". I'll leave it as an exercise for the reader to fathom its meaning. A good place to start is "mongrel" .