This is mostly a semantic issue.
When the sun is setting, the slope on the eastern side of a mountain is in shadow. It is, in fact, in the mountain's own shadow. In that context the shadow is the shade itself. Is night the Earth's shadow? You could, playing with language, state it that way, using both "night" and "shadow" as nouns. It would probably be more semantically accurate to make one or the other a verb, and say that night is the state of being in the Earth's shadow, or that night is the part of the earth that is being shaded.
Since the sillhouette of the Earth is cast, from any given perspective, by the Earth's horizon, any inhabitant of Earth is in the Earth's shadow at night.