"Everyone belongs to everyone else" is one of the mottoes of the New World. At the early age of seven or eight, children are engaged in erotic play in Chapter 3, and they are conditioned to abhor the concept of human parents who have actually co-habited and had a baby in a live-birth--the natural way, procreation, that is obsolete in the New World where babies are mass produced in a Bokanovsky's Process, "one of the major instruments of social stabilty."
Later in a dressing room in the London Hatchery, Fanny tells Lenina,
"It's such horribly bad form to go out with one man for seven months."
Close personal relationships are not really acceptable; better to have three or four men at different times: "...you ought to be a little more promiscuous...." Lenina replies, "Yes, every one belongs to every one else....I'll make the effort."
In the New World, there is no individuality. By having small children learn that one person is the same as another, erotic relations are insignificant. And, by being promiscuous, no one man matters to the women. So, there are no disruptive attachments.
The state considers marriage disgusting. When Bernard Marx and Lenina are about to go to the Savage Reservation, they learn from the warden that it is:
“About sixty thousand Indians and half-breeds. Absolute savages. Our inspectors occasionally visit. Otherwise, no communication whatever with the civilized world. Still preserve their repulsive habits and customs. Marriage, if you know what that is, my dear young lady; families. No conditioning, monstrous superstitions..."
Marriage is one of the "repulsive" customs there. It is also something an average person may not have heard of, as it has been abolished in this new world, along with biological parenthood. Now everyone is created in a test tube.
Intense personal relationships are strongly discouraged and seen as aberrant. Men and women are expected to flit sexually from one person to another rather than enter into committed relationships. Fanny's friend Lenina tells her:
"You know how strongly the D.H.C. objects to anything intense or long-drawn. Four months of Henry Foster, without having another man—why, he’d be furious if he knew.”
When Lenina says that she is sometimes disinclined to promiscuity, Fanny agrees, arguing, however that:
"One’s got to make the effort,” she said, sententiously, “one’s got to play the game. After all, every one belongs to every one else.”
Stability, as the Controller insists, is all-important, the bedrock of civilization. Marriage and committed personal relationships, the state believes, threaten stability. People are more committed to the state and its dictates if they don't have the competing pull of intense personal relationships. The state promotes superficial relationships and conditions people to like and accept them. Any residual emptiness can be dealt with through soma.