A doorway is a human interface between two spaces separated by a wall.
The only meaningful design attributes of a doorway is the shape (circle, triangle, quadrilateral, pentagon, etc) and the dimensions of that shape.
Let's say you're a doorway designer, and you're looking to design a general-purpose doorway that satisfies common use cases: allowing humans, dogs, groceries and strollers to move from one space into another.
tl;dr Design a rectangle: The bottom of your door probably needs to be flat, to let wide strollers pass through easily. It needs to allow a door to rotate about a fixed axis, which should be vertical so you don't have to lift anything at any point. To simplify the manufacturing process and reduce costs, it should have a minimum number of sides. Doors should also occupy a minimum of horizontal wall space (to allow ingress to small spaces) and accommodate tall individuals. The only logical design conclusion I see is that your interface is some sort of vertically-oriented rectangle, a little larger than the average human.
But let's say you want to copyright your doorway design. All you've done is chosen a shape (rectangle) and some dimensions (say, 32"x80"). That means you could theoretically own the exclusive rights to human-sized rectangular holes in walls. What are the odds that another doorway designer would have produced a similar, if not apparently identical, design? I'd guess astronomically high. And what are the odds that they did it before you? Again: quite high.
An API is just an interface between two pieces of software, and just as with real world interfaces, an API should be made as simply as possible such that they are intuitive, stable, and reliable. So, if it doesn't make sense to copyright something obvious like a generic doorway, how does it make sense to consider copyrighting an API?