using De Morgan's negation rule this is equivalent to
⇔ ∄ P ∈{X | X lives in Japan} : P is not named Sato
Since {X | X lives in Japan} = ∅ is the empty set, such a person P can by definition not exist. Which means, the first statement is true. If no person lives in Japan, that means every person living in Japan is named Sato.
For one, there is no legal requirement that a Japanese partner take the name of their foreign spouse (in fact, it's basically the exception to the rule that all married couples must have the same surname).