I understand that the customary angle of view is quoted along the diagonal. I don't know any body that uses the diagonal dimension as a primary measure for composing images. Why not give an angle of view on the horizontal dimension? Or both the horizontal and vertical?

Some manufactures do (I know Horseman does this for its SW series cameras), but for the most part the diagonal is the easy single figure. It works fairly well until the difference in aspect ratios becomes large. I am sure horizontal and vertical angle of view would be more common if there were more requests for it.