Not exactly, but sometimes. Speaking as a Brit living in Austria, I generally see three payment options available with a lot of online retail:

- normal credit/debit card, which can go through all sorts of different "middlemen", US or European based depending on the site

- "immediate" bank transfer, which is still integrated into the payment flow and provides confirmation straight away, and is usually handled by the Swedish/German middleman Sofort/Klarna

- suprisingly still very common, at least here in Austria, is the "send me an invoice" option, even for physical goods that are being shipped, then the customer makes a manual bank transfer after the goods have been received

Why offline? You go to your bank web page, login, enter the recipients account information (EBAN) and press transmit. Or you have that automated from SAP or a number of other software products.

SEPA is a thing and has basically 100% EU coverage. It’s cheap and well supported, and you don’t add any additional middlemen (you only have the banks that you’re using anyways)

Online banking still exists. That is, you log into your bank and specify the recipient's bank, account number (or IBAN), the amount, and the purpose of the transfer.