The issue is you order something, your significant other unpacks the thing and disposes the paper which includes the invoice to be paid. Only weeks later you get notified that you did not pay.
Could be different in your country, but here in Norway there wouldn't be an invoice in the box.
The whole idea is that payment is totally separate. Email invoice or separate invoice per post, unless you pay immediately when ordering.
The shop you ordered from does not know or care how the transaction is done between you and Klarna.
The shop you ordered from does not know or care how the transaction is done between you and Klarna.