Equity is on the right side of the balance sheet, yes, but not within the liability section (which identifies debt) but within the equity section (which identifies ownership). When a company is liquidated, the owners receive their share of the remaining assets because those assets are a shared property, not because the company owes them.
Anyhow, the property ledger argument still stands. Not all registries are liabilities, there are property registries such as a land registry or a custodian registry. The fact that ledger points to an intangible asset (stock in a custodian ledger, a trademark or Bitcoin) does not make the ledger a liability.
Current Central Bank fiat currencies are a paramount example of liabiilties because the central bank has to oblige itself to accept it to cancel debts. Without that self-obligation their issued currency would be obviously worthless.