First party lenses will be handled by either camera firmware or by the camera - for your lens (I'm in the Canon RF system heavily) you can do the "download to SD from canon.com" and the firmware update on the camera will recognize it as a lens firmware, not a camera firmware. And sometimes camera firmware will also bundle lens firmwares that get updated when you attach the lens (usually these are reserved for issues that have a potential to damage the lens or camera - like excessive hunting wearing the motor, etc., not 'nice refinements').

Yep. Sony lenses are the same. You can update the firmware on a sony lens by attaching it to a sony camera.