@SuperbattleshipYamato They get activated independently of each other, unless a country has multiple territories (Spain has Western Sahara; Portugal has Angola, Mozambique, and Guinea; and Mongolia has 6 constituent territories.)
So if you attack a nation, you only need to fight its territories and ships. If you failed or opted not to take or destroy all of them, the remaining must be placed into the hands of an opposing power (similar to how neutrals in 1914 worked).
So if you attack Portugal as the U.S., the Axis will mobilize Portuguese colonial troops and receive income and control over all Portuguese territories and ships. You can pick Italy or Germany or Japan or even Romania if you’re playing with minor powers. But you must represent that nation with only one power: you can’t split them up.
But if you attack Saudia Arabia or any other neutral with only one territory, nothing else gets affected. They’re independent now, and typically have a decent military.