The merchant typically uses stripe or adyen or whatever (mollie has a cute name!), a payment service provider or PSP.

The PSP looks at what methods the merchant wants to accept, which methods the user could potentially be using (based on e.g. country by geo IP or some delivery location) and show the relevant icons.

EU users will see schemes like wero or Przelewy24, Japanese customers will see 'konbini' among the icons, and US users may only see credit cards, Apple Pay and Affirm. There are TONS of payment services. Stripe lists 123 of them.

The merchant will want to exclude methods that have high costs (for themselves), maybe they also care about their customers not getting into debt (so no buy-now-pay-later or credit), and some payment methods have higher rates of disputes/chargebacks (e.g. Amex).

In general, most merchants will want to offer as many methods as possible to prevent consumers who have a preference (this week) for using account A over account B from bouncing.

Yes at the pay page there's a 'pay by X' list of options, you choose it.

You then typically have two choices: scan QR with your phone or login to your bank.

I normally open my bank app on my phone which signs in via my face (iPhone), I then press the scan button (first screen), point at my phone at the screen to read the QR code, the transaction pops up on my phone, I press confirm and again it signs via my face. Then you're done.

If you were shopping on the phone it's even simpler of course as the pay button opens up the transaction in your bank app right away, but typically I shop on my laptop after research.

I've had this for almost 20 years by the way in the Netherlands, but now it's pivoted to the EU standard.

Go to store, select "pay by Wero". Get redirected to payment screen where I select my bank. My bank just shows a QR code that I scan with my bank app, authenticate the usual way, redirect back to store, job done.

In Portugal there's often a "MB Way" payment option - which AFAICT is the PT version of these other systems.