It is worth noting that bodybuilders are basically training themselves to store excess protein calories in the form of hypertrophied muscles. There are 'non-active' forms of protein in the body, collagen for instance, that could be expanded as a means of excess protein storage, I think this would basically replicate excess adipose tissue in that you could store large amounts of collagen in the same areas you do fat. It is important to realize that abnormal protein build-up can be fatal, like in the case of amyloidosis or kidney damage. Adipose tissue is accumulated in specialized adipocytes, so you would probably need "collagenocytes" or "proteinocytes" to store excess protein safely and start the metabolic breakdown (into glucose) when necessary. You can probably do the same thing with glucose.
As to why we don't already have this capability, there are probably advantages to just having large scale fat storage instead of protein (outside of muscle hypertrophy) and glucose/glycogen. Temperature regulation, vitamin storage, water requirements, dietary availability, and ease of utilization all probably contribute. The obesity we see today is probably unique to just the last few millennia with easy access to high carb diets provided by agriculture. Caloric excess from plant based sources was probably a rare thing with human cultures prior to that innovation (versus gorging on fat and protein from a large animal kill).