I measure all kinds of shapes all the time. It really depends on what you plan on doing with the information after you have it.
I mean, the image itself is the ultimate description of the shape. Or you could give a list of (x,y) coordinates of the black lines. Or you could try to fit the black lines to some smooth shapes and give the coefficients of that shape and the residuals. What you measure really depends on what you want to do next. Let's say you fit the top of the shape cross section to a parabola. OK, but then what? Will that be useful to you? If so, fine. If not, what would be better? Saying your "main goal is to reduce bias in shape definition" really doesn't answer the question. I don't even know what you would consider bias. When you say you want to "quantify and compare them and then classify them", quantify how? By polynomial coefficients, for example you want to know if there are two classes of boat keels: U-shaped and V-shaped?
Have you tried to classify them into known classes using Deep Learning and transfer learning with AlexNet or some other network? This could be the way to go if you can label images with ground truth about what class each image belongs to, and you have dozens or thousands of images to train the network with.
