Describing system in terms of 2 variables vs natural variables

I often see a function's differential expressed in terms of convenient partial derivatives eg
dU=(dU/dT) dT + (dU/dV) dV

And I've seen it written that "any system is uniquely specified by two parameters, such as pressure and volume, or perhaps pressure and temperature"

But then what's the deal natural variables? What's so "natural"/good about them if any pair will do? By natural I mean that I've seen the natural variables for gibb's energy as pressure and temperature.