As a preschool teacher I can think of absolutely no reason I would need to know if a child is adopted.
I need to know who the legal guardian is and who has permission to pick the child up from school. We have to see an original birth certificate in order to register the child. If there is some bizarre situation, we might need to see a custody decree, although I don't think it's ever happened at our school. We have been shown restraining orders against non-custodial parents regarding pick-up privileges.
We do ask who the child lives with though, including siblings or "extended" family.
I actually think asking if a child is adopted is a violation of privacy. Many parents (although fewer in this day and age) still don't wish their child to know that they are not "natural" children. (I much prefer biological. All children are natural.)
How is it a violation of privacy if you are already asking for their birth certificate? It will show the birth parents--wouldn't you wonder why those names are different then the parents signing the child up for class--thus needing the adoption information?