Ever see The Parent Trap? Mom and dad get a divorce and each takes one twin, so the girls grow up in very different environments and have very different personalities as a result. There aren't twins in this Mansfield Park, but Fanny and her siblings and cousins come close. Family bonds are stretched over different circumstances and incomes. The novel raises the big nature vs. nurture question. Are people's personalities based on family genetics (nature) or the circumstances in which they are raised (nurture)? Are the characters that end up revealing some serious character flaws born that way, or is it a result of their parenting? In addition, most of the families we see aren't very happy ones. Blood may be thicker than water, but it isn't thicker than things like income or clashing personalities in this novel.
Mrs. Norris is the person most to blame for how the Bertram girls turned out.
We see kids both with and without parents here. Ultimately the book suggests that having parents can be just as problematic as not having parents, in terms of shaping character.