I think I know what's going on. Your description of "eating" chicken with kidneys combined with the picture you posted to demonstrate the kidneys....I think you are mistaking the tissue that is on either side of the spine for the kidneys. The kidney is the shape and color of a red kidney bean. In a chicken, it is also about the size of a bean. However in addition to the liver and kidney, there is tissue of the a similar color, that is located on either side of the spine. It is irregular in shape because it follows the contours of the spine. I believe that is what is pictured in your photo above and that would make more sense that you are seeing it when you eat the bird, as that is typically left in the cavity, while the kidney and liver are removed along with the intestines, heart and lungs.
I am very thorough when I clean a bird, as I want to make use of every single piece of it. Therefore, I keep the kidney and when I have enough of them, I use them to make a steak and kidney pie. I save the liver and we cook that up and eat it as liver and onions. We cook the testicles, heart and gizzard along with the bird. The intestines are fed to the animals. Because I carefully take apart every part of the contents of the bird's cavity, if there were more than one kidney, I would have been happy to add it to my baggie of kidneys, since with only one, it takes a long time to save up enough of them for a pie