Why was the punishment of "Nine Familial Exterminations" used often as a systemic punishment in China, but not anywhere else?

It's guilt by association.

First, Chinese society was organized along the lines of filial piety - unfilial acts towards parents and/or elder family members or the killing of parents and/or family elders was one of the "Ten Great Crimes" and punishable by death (this was codified in both Ming and Qing law). Because they were bonded by filial piety, families were supposed to remain strictly loyal to each other, and thus shared collective responsibility. Under this logic, a criminal's family would be most likely to shelter and aid him, so to the authorities, they were all guilty.

Second, Confucius said, "The silent treasuring up of knowledge; learning without satiety; and instructing others without being wearied - which one of these things belongs to me?" (Analects, Shu Er chapter) From this, we can see that Confucius believed that a good teacher was just as important as a good student. If a student performs poorly, then it reflects poorly on the teacher because it showed that the teacher failed his duty. Confucian society was one run under mutual obligations. The parents raise the child and the child in turn supports his parents. For example, the three year mourning ritual for parents was stipulated because for the first three years of a child's life, he is in the arms of his parents (Liji, San Nian Wen chapter). So similar to teachers, if a son behaves badly, then it reflects poorly on the parents because it showed that the parents failed to instruct the son properly. Under Confucian logic, a man would not rebel or commit treason if his parents had taught him well, so they bore responsibility for his actions.

Third, and most importantly, exterminating a whole family was seen as the ultimate punishment because it effectively doomed an entire clan to extinction. It was seen as disgraceful - the executed members would have no face in front of their ancestors. Mencius said, "不孝有三，無後為大" or, "There are three types of unfilial [behavior], the first being no [male] heirs." A son could not produce an heir was seen as unfilial towards his parents, so imagine how disgraceful it would be if that son lead to his entire clan's destruction. This type of punishment served as a deterrent to prevent people from rebelling or committing treason precisely because it was so harsh.

Interesting thing to note is that killing the entire clan as punishment was not relegated to just China, Joseon Korean had it as well, since they followed the Ming law, and I believe Vietnam did too (I may be wrong here since I'm not familiar with Vietnamese legal history). Also, the term "Nine Familial Exterminations" (滅九族，誅九族，誅族) never appears in the Ming and Qing law code. The code just lists the relatives who would be killed, and it's not just relegated to family members. Anyone living with the criminal or the criminal's family would also be killed, even if they have a different surname and were not in that person's family registers. Only those over the age of sixteen were to be killed, anyone under that age would become slaves or exiled, so the family is never really "exterminated". Of course, this was what the law said, whether or not it was actually followed to the letter is something else.

There was also a “Ten Familial Extermination”, used only once in China's entire history by the Yongle Emperor on an official named Fang Xiaoru. The "tenth family" was his students and acquaintances.