Debt is the liability evidenced with loan contracts and accounting books in the banking system.
Exactly, it's a liability. The asset which is promised to be paid is nothing but fiat currency. Fiat currency is nothing more than "numbers" whose power is forcefully indicated by the government. When I own $100, I only own the papers that say so.
You fail, miserably, to understand that the same happens with Bitcoin, but there's no authority that enforces its usage.