Yes but you need a social ID that is given specifically for that purpose and you need to be present here in Iceland to get it. Fairly standard I would guess.
"social ID" = "kennitalia" ?!?
Yes. "Kennitala" is your birthday example 12 Month example 05 and year example 93 than 4 extra digits ending with the decate you were born so standard kt =120593-2549
So I show up in some official place in Rejkjavik and request a kennitala (as a foreigner) for opening a bank account and they will issue one and put it into the database with certain properties that will allow me to use it for opening an account with one of your banks?
I think the application goes through the bank, the bank makes a special application for you for the purpose to create a bank account. You of course need to show a passport fill out and sign a money laundry statement etc. and I think you will then need to revisit the bank a few days later when the ID is ready and you can finish the process.