Does paying extra money on your loans count as savings?
Depends on the loan, if it is additional principal on your mortgage then yes, if it is paying extra for a car loan then no.
Interesting differentiation... since money is so fungible, how would you argue this point?
Paying towards principal on your mortgage is savings because it increases your equity in home (The interest portion of your mortgage payment is expense). An automobile is a depreciating asset so any payment toward an auto loan is an expense.
I agree. Much like buying a paper asset, you can hold something appraised at a certain value (gold bars, paintaings, real estate) or hold that value's worth in cash.
If you want to get technical you could consider the net of car loan payments after interest and depreciation savings because you can in theory sell the car after payments and transfer from a physical holding back to cash.
edit: A mathematically equivalent way is to deduct the whole cost of payments as an expense when the expense occurs. Then, when the sale occurs, I count that as income. I do this because there is a great deal of uncertainty in what the actual unrealized value is of a car (you could total it &sell it to insurance, sell to a scrapper, sell to a private buyer, sell to a dealership and all come with a different valuation).