That's an idea. It made me think of philosopher John Rawls' concept of the Veil of Ignorance: what would we all want the world to look like if our place in it was up to chance? How would we write the law of society if we had no way of knowing our place in it?
And that reminds me of Plato's Republic, in which children are separated from the parents and then brought up by others. The intention being to create ties to the community rather than parental lineage, so that the whole community is one big family.
People would rebel with tremendous violence against it. Either you would have a total revolution, or a dual system, where the elites had a separate set of rules for themselves. Probably the latter.
I don't know why but I have a feeling women would be more against it than men.
I'm lead to believe the bond that a mother has for the child that grew inside her for 9 months is something a man can never truly understand. I just asked my wife what she'd do if she found out our 3yr-old daughter wasn't biologically ours. She said she'd keep this child AND fight to get our biological one as well. I said, as painful as it'd be, I'd prefer to give up our current child and get our biological one....
There add numerous Sci-Fi novels and short stories that this pops up in. One I can think of off the top of my head is The Giver where children are carried by women who have that job and are then assigned to a family unit.
I remember a great science fiction novel, Dancers in the afterglow by Jack L Chalker, (well, it was great when I was in Grade 6) that had people meeting and randomly shuffling their own personalities/experiences with someone else after reaching adulthood. And then, after they had memorized and picked up the new personality, doing it again with someone else.
That's almost like what happens in the sci-fi series Farscape with the peacekeepers. They aren't ever supposed to know their parents, instead they are all raised in orphanages.