To elaborate upon Roman Tunkel’s important information, the two “kinds” of Günzburg (Cyrillic: Гинзбург, Гинцбург, Yiddish: גינצבורג, גינזבורג, Ginzburg, Gintsburg) in Russia both were Jewish.
In the first case, they were the descendants of one German Jewish family, some of whose members had migrated to Russia by the 1600s... and in the second case, they were from unrelated Jewish families in Russia who early in the emancipation period, when the Jews were ordered by the government to adopt family names, seemingly chose the name Günzburg because it was respected and/or sounded pleasing to them.
Indeed, apparently according to David Maggid, "Toledot Mishpechoth Gintzburg," Pg. 239, St. Petersburg, 1899, there even was a lawsuit instituted by Baer Günzburg of Grodno against a Jewish family of that city who had adopted the same name under the decree of 1804. The court sustained the right of Jewish families to adopt any name they chose, and the number of Günzburg families accordingly was said to have increased.
Usually cited as the progenitor of this family is Simon Günzburg zur Gemse, about whom the Jewish Encyclopedia, 1906 wrote:
“German scholar; communal worker; born at Günzburg, Bavaria, 1506; died at Burgau Jan. 9, 1585. He was the first who adopted and transmitted to his descendants the name "Günzburg" as a family name. He was a rich merchant, and traveled around in Germany and Poland in the interests of his business. He was also a great Talmudist, and had some knowledge of secular sciences. It is probably owing to these facts that Simon Günzburg is variously described by different historians. Albertrandy, quoted by Sternberg ("Gesch. der Juden in Polen," p. 148), says: "Simon, also called Selig Günzburg, was known as a celebrated architect and geometer. He wrote many works, and was the head of the rabbinate and yeshibah." It seems that Albertrandy confused Simon Günzburg with the physician Selig Günzburg of Slutsk. Czacki cites him as the court physician of King Sigismund August and chief of the community of Posen (Grätz, "Gesch." ix. 448). But Simon Günzburg never settled at Posen. His residence was first at Günzburg, where, he built a synagogue and established a cemetery; and then he settled at Burgau, a neighboring town. There also he worked for the welfare of the community, for which reason his name is commemorated in a special prayer.”
According to sources I believe to be reliable, his ancestors are documented to be:
1 Meir Zurich d. 1345
2 Baruch ben Meir Zurich b. 1319 d. 1382
m. Guetlin - Kalonoymos b. 1328 d. 1417
[daughter of Smoe (Samuel) ben Baruch and Name Unknown]
3 Lemlin ben Baruch b. 1355 Burgau (D) d. 1410
m. Zuerlin Unknown b. 1360 d. 1436
[daughter of Baruch ben Meir Zurich and Guetlin - Kalonoymos]
4 Falk ben Lemlin b. 1390 Augsburg, Bavaria (D) d. 1465 Ulm, Baden-Württemberg (D)
m. Name Unknown
5 Schmuel (Samuel) ben Falk b. 1418 Augsburg (D) d. 1478
m. Name Unknown
6 Jechiel ben Schmuel Porto b. 1445 d. 1505
m. Frimes bat Aharon ben Schmuel
7 Elieser Ullmo - Günzburg b. 1477 (P) d. 1544
m. Simche Unknown
8 Simon \ Schimon \ "Seligman" Ulmo - Günzburg b. 1506 Günsburg, Bayern (D) d. 9 Jan 1585 Burgau,
m. Hannele / Händel bat R. Isaak Linz m. ABT 1550 d. 1 Nov 1593
[daughter of Rabbi Isaak Linz ben Eljakim - Linz and Tserlin Unknown]
This indicates the family ‘originated’ in what now is Switzerland; and subsequent migrations ultimately brought some of its descendants from the city of Ulm on the Bavarian border to the town of Günzburg in Bavaria. It’s for this reason, Simon Günzburg and several of his immediate descendants, sometimes called themselves "Ulma-Günzburg".