Thank you for a very good question and one that's confusing to many people. In the world of food science, there is one very large umbrella category that includes both cholesterol and fat. That category is called "lipids." Unfortunately, this word isn't used much in everyday conversation. "Lipids" are defined as organic compounds that do not dissolve in water and that include fats, oils, waxes, sterols (including cholesterol), and triglycerides. Even though "lipids" is the correct scientific term for describing both fats and cholesterol, it's fairly common for people to say "fats" when they actually mean "lipids." So from this perspective, we think it would be fine to say that cholesterol is a type of fat.
Cholesterol is a very unique type of fat, however. Many types of fats have a fairly simple chemistry. For example, fatty acids (one very important kind of fat) are basically straight chemical chains. Cholesterol is more chemically complicated. Not only does it have chemical rings in its structure, but these ring structures have to occur in a very particular configuration. It's fairly complicated for our bodies to make this unique type of fat.
In a practical, dietary sense, there is one very basic difference between cholesterol and other fats. Most of the time when we talk about fat in food, we are talking about fairly large amounts of food parts that contain large amounts of calories. It's not uncommon for a fast food meal to contain over 50 grams of fat. These 50 grams of fat provide about 450 calories. This situation is very different from cholesterol. We almost never consume a food that has more than 1 gram of cholesterol, and we never get any significant number of calories from cholesterol. So in this respect, cholesterol is very different that other types of food fat. However, just because we don't need to worry about cholesterol in terms of excess grams and excess calories doesn't mean it isn't a type of fat with potential health risk. Excess intake of cholesterol can be a health problem in terms of our cardiovascular health, even though we now know that there is no simple relationship between dietary cholesterol and risk of cardiovascular disease, and that it takes more dietary cholesterol than we once thought to pose significant problems.