Egg sushi is seldom ordered or hardly even noticed today. It is a traditional Edo style sushi which dates back to the earliest days and unfortunately most of its tradition has been forgotten. It is usually ordered last and treated similar to as a dessert. It is cooked slightly sweet and acts as an appetite suppressant.
Egg sushi used to be a big deal in the old days and is nothing compared to what you typically see in sushi restaurants today. They consisted of ground white fish (halibut, red snapper..etc), reduced rice wine, sweet rice wine, bonito broth, and grated yamaimo which is a potato root local to Japan. The omelets took about an hour to cook both sides on very low heat. This made the omelet very fluffy and the texture becomes similar to a sponge cake, while the ground fish added deeper flavor to the omelet. The omelet were made thinner and made into nigiri sushi with the now seldom practiced hontegaeshi method so the egg sushi sticked to the rice without using a belt of nori to fasten the egg to the sushi rice(see typical egg sushi image).
Recent egg sushi are made as thick omelets, and consist of egg, bonito broth, and sugar. Some sushi restaurants still make the old fashion egg sushi (left image), but they are thicker, and are not made into the jigami style(left image) which requires more skill.