Are all fatty acids monocarboxylic acids?
You are asking two separate questions.   

Not all fatty acids are typically found in fats and oils (common name for storage lipids, which is also somewhat a common name...)  Identifying fatty acids that have an additional charged tail does not inherently mean you'll find them in your fats and oils - fatty acids do have other roles. There may be exceptions, but they're not going to be common. 

From what information you have given us, the larger question your teacher is driving you towards is to think about what the physical nature of fats and oils must be given their biophysical properties and how that correlates with their biological role, and secondly what the physical nature of fatty acids would be that drives those properties.  

Considered macroscopically, your storage lipids will remain neutral, and pack together. If you added charge to your storage lipids in any significant amounts, you would turn them into something that behaves like a membrane lipid.