Also, the Herbert Hoover reference in the show's opening theme wasn't exactly accurate either. He's not a Hoover Republican (he would have been about 5 years old when Hoover became president anyway) or full-fledged anti-New Dealer. He was a staunch union man. And his views on FDR were somewhat more nuanced - he bashed FDR more to annoy Edith's liberal cousin Maude than anything else. In this exchange Edith states that Archie's family was for Roosevelt, to which Archie replies that was for two terms. And there's actually an element of truth to that: while FDR enjoyed near-universal Jewish support throughout his presidency, support among Irish and Italians actually dropped off quite a bit in 1940 and 1944 (the Irish because of his pro-British stance and the Italians because of the "stab in the back" speech about Mussolini).
https://www.youtube.com/watch?v=10CRI7w_vIMOf course you can't expect TV shows to be 100% consistent.
Archie Bunker is "not Irish" in the way that the Costanzas on Seinfeld are "not Jewish."