>I haven't yet found a way for a mobile app or web page to explicitly signal to the device that the passkey to be created should live in $password_manager and not whatever built-in/on-device key-store exists.

On Android 17 (on Pixel) you can select the password service under Settings -> Passwords and passkeys -> Preferred service.

If you have an alternative password manager installed, it will be listed there along with Google's own password manager. iOS has a similar setting but I don't know exactly where off the top of my head.