Are city school boards, responsible for elementary to high schools, the same boards for colleges/universities?

I am not aware of anyplace where the same board or department is in charge of colleges and universities as primary and secondary schools. In most states, there is a board of regents or chancellors that governs state-run colleges and unversities and accredits private colleges and universities, and each institution has its own board of trustees for its own governance. There would not be anything at the city level - in an area with many colleges and universities there might be some type of consortium that allows students to transfer credit easily or sharing of resources such as transportation, dining halls, etc., but it would not fill the same role as a school board or board of regents. If I were you, I would start by going to the state government webite of the state you are interested in and searching for colleges and universities - that should direct you to the appropriate entity.