Indeed as you say, a subgroup of the largest Swedish private banks own the ID system in Sweden - for profit, and without any serious democratic oversight.
Edit: I forgot to add that the system allows these private banks to see into almost every aspect of a person’s life: where they shop, where they are, who shares their household and so on. Almost every aspect of a Swede’s life can and is tracked by this system.
Every time someone identifies themselves with this system, it costs the retail merchant or service a non-trivial amount of money. Because it’s effectively a private monopoly, that price is set by the banks, and often involves a lot of secret horse-trading behind closed doors (I’ve been involved with some aspects of this in the past).
The secret negotiations also include terms that are not open to public scrutiny. One example, is that the merchant or service isn’t allowed to blame BankID for any problems such as downtime or any other technical problems.
btw I’m curious how you get all your receipts digitally. There are some services such as Kivra in Sweden, but they definitely don’t cover all stores.