I'm pretty sure his name isn't "Gustav the second, the second of his name."

Having a Latin-ized name isn't particularly notable, especially for royalty who are trying to compare themselves or their reign to the Roman Empire (or barring that, the historians trying to mark a connection). Latin was also the language of science (which was a much broader field than we think of it today) so we see the names of ancient non-European based figues being Latinized both for convenience (as their native forms might be hard to say) and also as a comparison with Classical Heritage: for example: Avicenna (for ibn-Sina), Confucius (for Kong-Tzu), or Copernicus (for Kopernik). A good example (though Francized, not Latinized) would be Charlemagne, which is the most common and acceptable form of the man's name in English, whereas the translation "Charles the Great" is largely unknown (on the other hand, the form "Karl der Grosse" is the acceptable form in Germany, which is the literal translation).