Rothbard argued that children cannot necessarily be owned, but that people can owner a “guardianship” for them until the child “homesteads” his/her free destiny (an example of this would be voluntarily moving out of the house). Generally a system like this would tend towards a system where the best-fit parents are caring for more and more children.
I tend to view being born into a family unit as feudal, especially considering that a great number of families nowadays are extremely dysfunctional, abusive, etc and greatly shape the psyche of the child as he/she grows up and matures psychologically and such. However, it is the state that props up this form of feudalism (just as it propped/propped all the other forms of feudalism in history) – by forbidding child labor, enacting minimum wage legislation and rent control/regulations, controlling and thus inflating the costs of schooling, etc, it makes it nearly IMPOSSIBLE for a child to escape from a psychologically and/or physically abusive situation.