Tartary, more than an actual territory, is a term by which West Europeans referred to a vast part of Central Asia, that covers parts of the modern Russia, China, Mongolia, Kazakhstan, etc. This region is mentioned once during the novel, when Victor mentions it as one of the many places of the world that jhe has travelled to while chasing "the Monster", after apparently hearing him laugh on the cemetery where William, Elizabeth and his father were buried (something that made Victor think that the monster was William's murderer).

Because of the distance and the difficult access , the region of Tartary, its cultures, geography and most of its history, remain unkown for most Europeans until 19th Century (after the writing of the novel). With most of their knowledge being limited to what they know of ancient empires like the Mongolian, and probably some records from the surrounding regions (India, China, Ottoman empire and Persia).

Tartary may have awaken the amusement and speculation 0of europeans at the time of the writing of the novel, but the author doesn't give many attention to it and just mention it as a remote region at which Victor Frankenstein was for  a short time chasing the creature. Honestly, while reading the text, I got the impression that when they say that he chase it through  Tartary, it reminds me of the expression "chasing/following him to ther end of the world", as it may have been an equivalent expression for Europeans of those times.

Sources:

Connell, Charles W. (2016). Ryan, James D. (ed.). "Western Views of the Origin of the 'Tartars': An Example of the Influence of Myth in the Second Half of the Thirteenth Century". The Spiritual Expansion of Medieval Latin Christendom: The Asian Missions. The Expansion of Latin Europe, 1000–1500. New York: Routledge: 103–125. ISBN 978-0754659570.

Gorshenina, Svetlana (2014). L'invention de l'Asie centrale: histoire du concept de la Tartarie à l'Eurasie. Droz. ISBN 978-2600017886. ISSN 2235-1353.

PUblic Domain image get from: File:1806 Cary Map of Tartary or Central Asia - Geographicus - Tartary-cary-1806.jpg - Wikimedia Commons