I would say that Atwood's "The Handmaids Tale" is as much Sci-Fi as similar low-tech dystopias, e.g. "Parable of the Sower" (Octavia Butler), "A Canticle for Leibowitz" (Walter Miller).
But the difference is that Atwood was, when she wrote "The Handmaids Tale" an author but not a Sci-Fi author. Whereas Butler and Miller were Sci-Fi authors.
And yes, "is this book sci-fi?" is a fuzzy category, and of no great importance.
I respect Margaret Atwood's opinion here but I don't quite hold it myself. More so about Oryx and Crake and sequels.
By "IDK if "The Testaments" qualifies though" I mean that I have not read it, just "The Handmaids Tale" so I'm not going to opine on something that I don't know about.
If it helps, The Testaments is firmly in the same genre as The Handmaid’s Tale, whichever genre one wishes to assign to either.