This is an interesting question to consider because I have always thought the dystopian world presented to us in Brave New World to be a lot softer and less harsh than the grim dystopian futures of novels such as Fahrenheit 451. After all, if you rebel, you only get exiled, rather than brutally tortured or killed. However, one of the similarities that we can draw between them is how the future societies these novels present us with clearly try to dismantle the bonds of love and affection in institutions such as marriage and family.
Note how Brave New World has managed to do this and why they feel that getting rid of parenting altogether is a good idea:
What with mothers an dlovers, what with the prohibitions they were not conditioned to obey, what with the temptations and the lonely remorses, what with all the diseases and the endless isolating pain, what with the uncertainties and the poverty--they were forced to feel strongly. And feeling strongly (and strongly, what was more, in solitude, in hopelessly individual iolation), how could they be stable?
In this novel then, institutions such as marriage and the family have been dismantled because they involve unpredictable and uncontrollable emotions that go against the stability that the regime is so desperate to achieve.
In Fahrenheit 451, we can see that what comes between Mildred and Montag's marriage is the "family" that is beamed onto the walls that Mildred spends so much time watching. We see that in this world, media has deliberately interfered with and become more important than the bonds of love and affection that are so intrinsic to human relations. Note what Montag says about his relationship with Mildred:
Well, wasn't there a wall between him and Mildred, when you came down to it? Literally not just one wall but, so far, three! And expensive, too! And the uncles, the aunts, the cousins, the nieces, the nephews, that lived in those walls, the gibbering pack of tree apes that said nothing, nothing, nothing and said it loud, loud, loud.
We see how family has been superseded by a shallow, ersatz imitation transmitted by media that changes human relations and attacks concepts such as love and loyalty. For Mildred, her "family" is clearly more important to her now than her actual family.
Both novels are thus similar in the way that the future world that they present us with attack or dismantle institutions such as marriage and family.