Fermentation can be defined as the chemical conversion of carbohydrates into alcohols or acids. Here sugar is the main substrate which is converted to alcohol. Can such a process happen in proteins and fats also?
As far as I know, no. Fatty acids are broken down into acetyl-CoA, which has to be metabolized through the Krebs cycle. Since the Krebs cycle cannot operate without oxygen, fatty acids cannot be burned for energy under anaerobic conditions. Similarly, many amino acids are broken down to intermediates of the Krebs cycle (e.g. acetyl-CoA, α-ketoglutarate), which also cannot be metabolized without oxygen. Some amino acids can be broken down into oxaloacetate or pyruvate which could be used to fuel fermentation, though I'm not sure whether these steps occur under anaerobic conditions.
It would be better to define fermentation as the facilitation of a chemical reaction involving organic compounds in the absence of oxygen. The fermentation of carbohydrates certainly fits under this definition.
It's possible to ferment amino acids with one of the amino acids acting as the electron donor and one acting as the electron receptor. See Stickland Fermentation. I imagine anaerobic bacteria do this frequently.