I may add that the time period that Frankenstein happens in should be considered.
As I remember, one of the characters returns to Geneva too late in the day and finds the city gates are locked for the night and they can't get in.
The fortifications of Geneva were demolished starting in 1849.
Victor Frankenstein studies medicine at the famed University of Ingolstadt in Bavaria.
A university that was relocated from Ingolstadt to Landshut in 1800.
I may add that the frame story which tells how a polar explorer encounters both the monster and Victor Frankenstein, and Victor tells his story, the bulk of the novel, to the explorer, has a partial date.
17__, if I remember correctly. So all of Victor Frankenstein's previous life and relationship with the monster happen before he meets the explorer in the Arctic, in the year 17__.
These events could be happening during the French Revolutionary Wars starting in 1792, but so far as I remember there is no mention of any wars during the events. Thus Victor Frankenstein probably goes from Geneva to study at the University of Ingolstadt, Electorate of Bavaria, and returns to Geneva at a time when Switzerland and southern Germany are experiencing peace.
Thus Frankenstein happens during the 18th century, and during a time of apparent peace between the numerous 18th century wars, so probably before 1792. But considering the relatively advanced ideas mentioned during the novel, it probably doesn't happen many years before 1792.
Geneva was a city state in the Kingdom of Arles or Burgundy in the Holy Roman Empire and allied with some of the states in the Old Swiss Confederation. As such the Republic of Geneva was a small semi independent realm such as was common in Europe in those days. This was before the rise of nationalism in the 19th century, so a person's sense of ethnic identity was not necessarily that of a broad linguistic nationality. The citizens, subjects, and inhabitants of small semi independent or fully independent states considered their identity to be citizens of those small states even if there were millions of persons outside of their small states who spoke the same languages.
When Victor Frankenstein says:
I AM by birth a Genevese; and my family is one of the most distinguished of that republic.
He means that he was born a Genevese citizen as a child of two Genevese citizens, no matter where he was physically born. Just as today an American citizen born when his parents were working in, for example, Germany, would say that he was born an American citizen despite being born in Germany.
In short, Victor Frankenstein was stating his nationality and citizenship, and by saying "by birth" in that sentence he merely means that he didn't emigrate to Geneva and become a nationalized citizen but was a citizen by right of his ancestry at the moment of his birth.