I think that it does make conceptual sense. For means and proportions, we want to compare the amount at which their values are apart. Which is why difference is used. For variances, the factor at which one is different from the other is important since it is the spread. Which is why ratio is used.
Someone told me here in another thread that a difference of variance does exist but I have tried searching and not seen any results. Can you just apply any operation to the multiple population measurements? I guess you could but they would probably not be worth of interest in studying.