About the study
The Zanzig name seems to occur first in the Stavenhagen parish region of pre-unification Mecklenburg-Schwerin, Germany. The earliest Zanzig or Zanzich names I've traced go to the 1700s and occur in both Mecklenburg-Schwerin and areas of Mecklenburg-Strelitz and Pomerania surrounding the border nearby Stavenhagen. Although the name occurs further west, in Parchim and Hannover among other places, in each case those people have ancestors in the Stavenhagen region. However, there are several families in that area that I haven't yet tied together--in fact, there are about 130 distinct family groups, including my own great-great-grandfather and great-great-grandmother who married and were both Zanzigs (and presumably were beyond 2nd cousins to be allowed to marry).
Zanzig and Zanzich are recognized variants, though Zanzich was replaced by Zanzig starting in the late 18th century. I don't believe there are any current Zanzich names exant. Zwanzig, though close, is not a variant but a recognized name in itself.
It is possible that the name traces to Danzig, now Poland, though this is uncertain.
The Zanzig/Zanzich name is so rare that we believe all people with the surname are related by blood.