This is not a school house and not a lighthouse. It is what is known as a “range light.” There are two lights, one at each end of the walkway in the photo. The other light (not shown in this photo) is near Baileys Harbor and when ships lineup the two lights, they have found the safe route into the harbor.
See my whole collection of Door County photos: https://greatrivers.smugmug.com/Galleries/Door-County-3/
https://greatrivers.smugmug.com/Galleries/Door-County-3/i-Hv6sT6f/A