This feature still works completely fine on GrapheneOS, but seems to require Google's credential service to be enabled in order to function when Vanadium is set as your default web browser. I had to go to the Settings app > Passwords, passkeys & accounts > enable "Google" in order to activate the feature last week.
I think the Play services dialog claimed that the passkey gets saved to your Google account, but when I checked my Google account settings later it didn't actually sync any passkey related to that.
The app uses a custom tab for this feature, and custom tabs share state with the default web browser. I didn't have to change any settings in Vanadium.
Funny thing, Vanadium now offered me to store the passkey in one of my password managers, or on a security key using the app HW Fido2 Provider. That didn't actually work though, so I'm assuming storing the passkey externally gets blocked by BankID.
To be clear, this feature is now called "BankID med biometri".