In Book II of the Politics, Aristotle dedicates several chapters to critiquing Plato’s notion of communism, or shared property, women, and children. As for property, he argues that Plato’s Republic would face the many well-known difficulties of any alleged communism: neglect of property, infighting, laziness, etc. And as for the common use of women, Aristotle complains that, without the constancy of housewives, households will be similarly neglected. However, it is his argument against the sharing of children that is the most interesting:
Again, how strange it is that Socrates, after having made the children common, shall hinder lovers from carnal intercourse only, but should permit love and familiarities between father and son or between brother and brother, than which nothing can be more unseemly, since even without them love of this sort is improper. How strange, too, to forbid intercourse for no other reason than the violence of pleasure, as thought the relationship of father and son or of brothers with one another made no difference.
I have to admit that I’d never considered this possibility until now. So basically (homosexual inclinations aside), children should not be shared in common because that might result in accidents like this: