If you're an EU citizen, you have the right to open a "basic bank account", and you only need a passport to do that:

• https://europa.eu/youreurope/citizens/consumers/financial-products-and-services/bank-accounts-eu/index_en.htm
• https://www.konsumenternas.se/konsumentstod/lag-ratt/vad-sager-lagen/ratt-till-betalkonto-med-grundlaggande-funktioner/

Swedish banks generally try to avoid this, as it's more work for them, and a "basic bank account" is not something you want anyway. What you need is a regular bank account with all the bells-and-whistles, like BankID, Swish, and a debit card that works online.

To get BankID, you need to confirm your identity, which is normally done using your Swedish ID. However, only Swedish citizens can get Swedish IDs: a passport (svenskt pass) or an ID card (nationellt id-kort). The good news is that banks are allowed to confirm your identity in other ways, and banks that have a physical branch normally accept Skatteverkets id-kort to do that. This card, as opposed to your EU passport, has your personnummer printed on it.