My stepftaher sent me to school under his last name without adopting me .Was he allowed to do this . What are the laws in this case .

The laws are those in the state where you live ( if in the US) and the local school board.

To put you in school, at least anywhere I am awhere of, they will need your birth certificate and you will have to be enrolled under your real name.

If not, you will have no real school record latter if someone was to do a background check.

If your school records are in one name, and that is not the name on your birth certificate, you need to notify the school.

Also unless adopted a stepfather can not actually enroll you in school, the parent with legal custody has to do that.