To get the typed login saved into the password manager, wrap the fields in an AutofillGroup and set onDisposeAction to cancel. The default tries to save whenever the screen is left, even after a failed sign in:
Also call it after a successful password reset, so the manager updates the saved entry.
On iOS the credential gets saved under your app's name. To tie it to your domain instead, so the same login also fills on your site in Safari, add the Associated Domains capability in Xcode with webcredentials:yourdomain.com, and serve this file at yourdomain.com/.well-known/apple-app-site-association with the application/json content type: