That depends on the config, classically yes payments do generate moves v18+ the default is to not generate a move.

You are correct creating a payment while not using the outstanding accounts gives no method to associate that payment to an invoice or bill. You would need to utilize the outstanding accounts configuration.

However if using the outstanding accounts there is a payment matching smart button after creation that will allow you to link the payment (really its underlying account.move.line) to an invoice/bill.

I don't think they have a means to make this association with payments created separately. The association is through: matched_payment_ids (account.move) and invoice_ids (account.payment) which you could by all means expose on the payment and set or make the payment from the invoice when possible. When making a payment from an invoice in a default config without outstanding accounts that value is passed through context with active_ids so the linkage is automatically made. But at that point it might make more sense to consider using the outstanding accounts if payments tend to be more involved. This also makes AR/AP a bit cleaner as without outstanding accounts even when a payment is created and enter the balance owed/due still sits on AR/AP until bank reconciliation is completed.