What? It doesn't mean this at all. You can have savings and still be underwater. If you have a 3.5% for your mortgage, but the market crashes after you bought at the peak, you can be underwater because your house lost value. And it would be foolish to pay down the loan rather than get a better rate for your savings.
Underwater mortgages are totally orthogonal concepts to savings.
> if you are paying a mortgage, a portion of that is a portion you are saving
I think I see your confusion. Yes, money saved into a home is an asset that you are building over time. However it's a non liquid asset, it's difficult to turn back into cash for, e.g., a surprise medical bill.
Repayment of a loan meets the strict mathematical definition of "savings" however it is typically excluded (as are, e.g., investments). For instance, the U.S. calculation of GDP does not consider repayment to be saving.
Savings are typically held in checking accounts, savings accounts, CDs, and money market accounts. Some people take a portion of their savings and invest them in higher risk items like stocks and bonds, but those are investments.
It's important to decouple the concepts of net worth (or even just worth) from the concept of "savings" because you can be taking actions that increase net worth (e.g. paying a mortgage) that don't increase one's savings. (Though, eventually, if you sell the home you could put any proceeds into savings.)