This may be true in some theoretical sense, but in practice, it is hard to believe. How would I go around to make the government accept my 20$ and give me some equivalent of that? What could government possibly give me for my 20$?
Dollar bills are for buying things. I could go to a store and buy a bottle of expensive wine for that bill. But I wouldn't say the store owes me.
That's a misunderstanding. A $20 bill is the government's liability, not the store's liability. You and a third party are using government liabilities as a currency. Every $20 bill is on the liability side of the central bank's balance sheet. Every entity accepts its own liabilities as a form of payment. Governments issue currency which they then must accept as payment for taxes. Banks issue credit which they then must accept as payment for debt service.
As an example, British pound notes have the wording "I promise to pay the bearer on demand the sum of five [ten/twenty/fifty] pounds", which dates back to a time when you could actually exchange the notes themselves for gold.