It’s not the buyer saying “I owe you”, but the issuer of the currency (actually, usually just the notes, coins are considered to have value). The first person/entity to get the note gave, or promised, the issuer (usually the central bank) something of value, and the issuer gave them a token (note) saying the bank owes the holder of that note a certain amount of value. The recipient can then trade that note freely, as can future recipients, in the knowledge a vendor will accept it for its face value. So, yes, you’re trading debt when you use money, but it’s the bank’s debt to the holder, not the debt of the buyer.

Typically the bank issue money when someone takes a loan, i.e. promises that they will pay the bank the value of the loan plus interest.