Certainly not in the sense that "we take away your children and raise them separated from you" but more in the sense of "it takes a village to raise a child". This can mean anything from extended family to patchwork to an active and engaged neighborhood to queer constellations of open relationship or poly or what ever. There is a quote from Thatcher "There is no such thing as society: there are individual men and women, and there are families."
Nuclear families, more than other forms of family and relating to each other, are isolating and making the people dependent on each other, most often making women financially dependent on men and men emotionally dependent on women. Abolishing the nuclear family doesn't mean that you can't live in a healthy monogamous relationship with a good connection to your kids. It means that you don't have to but can leave a toxic relationship and that your kids have other caregivers to complain about you and, if need, can leave. Or to live in other ways together that don't fit the model at all. To get back to Thatcher, it's not about taking away the little connection the individuals have but about strengthening the society, she denied exists.
I hope that helped. Sometimes I'm too much in my bubble to realize that implications aren't obvious. I specified "nuclear family" but I see that that's not enough. Thanks for pointing it out. Family Abolition is an interesting topic you can look deeper into if that interests you.
Anarchism is a broad term so I'm not going to generalize but many (if not most) anarchist schools of thought say we need a system that prevents rulers to emerge. It's not the absence of order but the absence of hierarchy.
By bullying the kid, they created a hierarchy and in a truly anarchist society, people would have stood up for the oppressed.
You should have said "read theory" and leave as any good leftist would have done.