While other grain-based foods (grits, cream of wheat, congee) are considered breakfast foods, the grains they're based on are also used in other foods that are traditionally served throughout the day.
So how did oats/oatmeal become a food so closely associated with breakfast, and generally not used for foods served during the rest of the day (oatmeal cookies?)
While oats have been cultivated since 6000BC, the ancient Greeks and Romans only used them as livestock feeds. However the Celts and some Germanic tribes consumed them for food (Pliny mentioned Germanic people ate oat porridge). At one point oats were eaten with any meal like other grains in the British Isles where oat bannocks were eaten by poorer people who could not afford wheat in Scotland, Ireland, Wales, and parts of Northern England as one of the most common bread for centuries. Oats were introduced to North America by early European explorers and they were cultivated in Virginia in the early 1600's. Scottish, Dutch, and other immigrants used them in their traditional porridge, puddings, and baked goods. Hannah Glasse's Art of Cookery Made Plain and Easy includes oats in recipes for cake, haggis, flummery, and hasty and other puddings. There were extensive recipes published for baked goods using oats included Scottish and English griddle oat cakes, muffins made from cold cooked oatmeal, and bread and biscuits, for which the oatmeal was usually mixed with flour, because on its own. In 19th century America, oats were considered health foods. They were recommended to invalids and served as hearty breakfast. Oatmeal became even closer associated with breakfast after rolled oats were developed by Quaker Mill Company in 1877. They are much faster and easier to cook than the traditional oat groats and were heavily promoted in the late 19th and early 20th century as a healthy and convenient breakfast (this coincides with the development of many other breakfast cereals at the time such as cornflakes, shredded wheat, Grape Nuts etc). At the same time other food made with oats became far less popular. Wheat flour became far more available due to increased productivity and the popularity of British style pudding made with oatmeal waned significantly as both the English and Americans expanded their food varieties beyond traditional British fares.
Sources:
Oxford Encyclopedia of Food and Drink in America by Andrew F. Smith
Oxford Companion to Food by Alan Davidson
A Second Handbook of Anglo-Saxon Food and Drink: Production and Distribution by Ann Hagen