In Wisconsin, it's in between.
Stepparents have the right to petition for visitation and, in rare instances, custody (for instance: if stepmom married a widower, and then he died, she could petition for custody and likely get it over grandparents, etc.).
Stepparents are also considered parents for the purpose of FMLA--definitely on the state level, and I believe federally as well. Stepparents can put stepkids on their health insurance, although sometimes they have to prove there's a court order that their spouse carry the kids' insurance and they have to carry the spouse, too.
Stepparents and stepkids can transfer a car between each other without paying a transfer tax here.
They also have the responsibility to care for the child in that a stepparent can be charged with neglect just the same as a parent can. Anyone--a babysitter, teacher, friend's parent--can be charged with neglect for, say, locking a kid in their care out of the house in the cold without shoes or a coat (basically, acute acts), but a stepparent can be charged if the children have inadequate food, clothing, or shelter.