Atwood in this dystopian classic paints a picture of a powerful theocracy that has taken away individual possession from its inhabitants just as it has stripped away freedom from those that dwell within its boundaries. This is true not just for the handmaids like Offred but also for those higher up in society who apparently have all the privileges and opportunities. Note the following quote said by Serena to Offred when she arrives at Serena's house and consider how it ironically uses the possessive adjective "my":
As for my husband, she said, he's just that. My husband. I want that to be perfectly clear. Till death do us part. It's final.
Serena even goes as far as to quote the marriage vow to Offred and repeats the phrase "my husband" twice in order to emphasise her ownership of her husband. However, the irony is that even though she is the Commander's Wife, the society in Gilead has dictated that her husband enjoys a sexual relationship with Offred, making any such notions of "ownership" questionable. Even though Serena is married to her husband, the use of a handmaid indicates that she cannot really claim full ownership of him. Even characters such as herself have to confront the fact that the society in Gilead has taken everything from them, meaning that the use of such possessive adjectives only exposes the intense desire to hang on to what society takes away.
I think that the most evident answer to this is that there is really no form of individual possession. Using the possessive mode of verbal recognition denies the reality that exists in the narrative. The totalitarian state in Gilead owns everything. Possession seems to rest with them. In matters of economic and political, it owns all. Its nature of being as form of power that has consolidated everything, it also seems to extend its possession towards how individuals in the state think with brainwashing center. The leadership of Gilead owns individual identity and punishes anyone for expressing it. There is very little notion of "my" or "mine." Women are essentially harvested for giving birth and those who cannot give birth find themselves essentially chained to a condition in which their lives are wasted away. In this, there is little else to suggest that anyone has any possession over anything else. Individuals are not able to articulate a condition of being in which life holds any sort of possession. The use of such possessive terms becomes ironic to an extent because few own anything of value since the state owns everything and it claims ultimate possession.