There are even visual similarities between the dress of women on the FLDS compound and in Atwood's imaginary Republic of Gilead: women's dress is strictly regulated, uniform and color-coded. In Handmaid's Tale, the color coding sorts women according to station, with Handmaids in red, Wives in blue, prepubescent Daughters in white, and servant Marthas in green; this picture taken of women and children exiting the FLDS compound seems to show them color-coded by age.
As pervasive as the similarities between the FLDS way of life and Handmaid's Tale are, though, I think there's a better analogue elsewhere in feminist science fiction. In Sheri S. Tepper's The Gate to Women's Country, the main character, Stavia, goes on an exploratory mission to try to find other areas of human settlement, and finds a polygynous enclave whose members forcibly abduct her and make her one of their leader's wives. I think this is a truer analogue structurally, because what makes Gilead oppressive is its omnipresence; each household is just a microcosm of the larger totalitarian state, and any woman who managed to escape her household would have to escape detection by the authorities, who in a police state are ubiquitous and practically omniscient. The FLDS patriarchs do not have the backing of an entire government and military; their power comes from their isolation. Women who have spent their entire lives inside the sect are practically helpless to escape it (of course, some do, but the odds are dramatically against them, which is why those who do escape are so noteworthy). They are uneducated, have no skills or prospects to survive on their own, have no friends or family outside the cult, and are kept ignorant of the resources and programs available to help them (FLDS children are taught to run from child-protective agents and other officials, and to lie whenever a stranger asks about where they live or whose child they are). The enclave in Gate to Women's Country is similarly isolated; the sect's members are a single family that has inbred for generations, taking the occasional captive from Women's Country's exploratory teams, but largely undisturbed by outsiders and unaware of any other human settlements existing. If women there wish to escape, they have nowhere to go.