Fats are mainly referred to triglycerides. In triglycerides, a glycerol molecule form three ester linkages with fatty acids.
I think it is also possible for a butane-1,2,3,4-tetraol molecule to form four ester linkages with fatty acids, or for a ethane-1,2-diol to form two ester linkages with fatty acids. But why they not considered a form of fats, or to make my question clearer, why should fat be triglycerides?
Indeed there's nothing stopping such molecules from existing. In fact it is possible to make dendrimeric polyol polyesters such as Olestra, where sucrose (a disaccharide containing eight free hydroxyl groups) has 6-8 hydroxyls esterified with long-chain fatty acids. Olestra (and presumably a number of other similar polyol polyesters) behaves much like regular triglyceride cooking oils, and it was used in the recent past in commercial products for human consumption.
However, these do not arise from natural biochemistry, and Olestra at least does not appear to be significantly metabolized by humans. As such, considering them to be fats may be technically correct by slightly extrapolating their chemistry and material properties, but biochemically speaking is probably a discouraged classification.
Ben Krasnow of Applied Science has a nice video with the story, properties and synthesis of Olestra, and he even fries some chips in it!
