Seems that what you've described is essentially equivalent to a pinhole camera. For both cameras, only a single ray of light goes from any single point on the object to its corresponding point on the image plane. In contrast, with a standard camera lens system, the lens captures multiple rays of light over its lens surface from any single given point on the object and directs all of those light rays to the same corresponding point on its image plane. That results in a great increase in light collection efficiency and shorter exposure times. In contrast, both your proposed "90-degree" camera and the pinhole camera are very 'slow' and inefficient camera systems since they only allow a single ray of light from any given point on the object to get to its image plane. All the other rays of light are effectively rejected by the cameras.
One significant difference between your proposed "90-degree" camera and a pinhole camera is that for your proposed camera the image size is guaranteed to be the same as the object size, whereas for a pinhole camera the ratio of the image size to the object size is equal to the ratio of the image-to-pinhole distance to the pinhole-to-object distance. If the pinhole is placed precisely midway between the object and the image plane, then the image size will be exactly the same as the object size (but inverted and with left-and-right switched).
Although your proposed "90-degree" camera and a pinhole camera are similar in the sense that they are both "single-ray" cameras, I think that a pinhole camera offers more flexibility. For your "90-degree camera", the only way to take a picture of a large object such as a mountain in a single exposure would be to build a camera as large as the mountain. There is no way to adjust the camera magnification. With a pinhole camera, on the other hand, you could take a picture of a mountain on a small piece of film by adjusting the image size by the method mentioned above.