Of course all fundamentalist oppressive branches of religions, be it Christianity, Islam, Judaism or whatever, converge to similar shapes.
Atwood was worried about trends in the US and imagined a world where the earliest conservative Puritan Christianity -- that she considers a foundational part of America -- made a comeback. So no "gender equality". The early Puritans didn't behave "like Islam", any similarities are a result of a convergent evolution of sorts; their sins are entirely their own: fundamentalist Christianity. Atwood said much, that these were faults of the West she was describing.
I'm all for Death of the Author, but the specific claim I refuted was about what Handmaid was intended to be about (not other valid interpretations) and only the author can answer that. And she did!