Well, it depends on exactly when the book takes place. Patagonia is southern Argentina and Chile. For much of Argentine and Chilean history, this area was not part of the state--think of the American west. There was a gradually expanding frontier that rapidly expanded in the last half of the 19th century. I was at a lecture once where they were discussing race relations in Latin America, and one of the speakers argued that Chile and Argentine were outliers because "they killed all the indigenous people" in the 19th century. He was being somewhat bombastic, of course (I think he himself was Argentine), but it's a history that looks more like than American West than the rest of South America. In Argentina, the rapid expansion kicks off with the 1872 Conquest of the Desert and in Chile, the conquests take place between roughly 1861-1883 (Chile was in a weird situation where they controlled both the north and the south of the current country, but Mapuche controlled the middle).

After the conquest, these very fertile lands help Chile and Argentine becoming major players in the world economy. In the early 20th century, Argentina's per capita income was comparable to Western Europe's (look at this article in the economist, especially the first chart). So if we're talking very late 19th, early 20th, living like a king in Patagonia would be living like a king indeed.

If we're talking about before the conquest, however, or the area just around the conquest, then the area had a reputation as a lawless, "savage" place that had yet to be cracked by white civilization. Orélie-Antoine de Tounens, for sample, was a French adventurer who declared himself "King of Araucanía and Patagonia" in 1860, as the Mapuche were preparing to fight the Chileans and Argentinians. He went unrecognized by the Chilean and Argentinians (but weirdly widely recognized by the Mapuche, even ones he didn't initially seek to "rule over" sought him out), and was deported by France by the Chileans in 1862 (he returned in 1869-1871 but ran out of money, voluntarily returning to France; he returned and was deported again in 1874 and yet again in 1876). By the time of the Chilean and Argentine conquest, most of the natives of the region had united in a single Mapuche culture (called the araucanization of Patagonia), and there was clear conflict between the Mapuche and Argentinian and Chilean ranchers who wanted their lands. So in this context, living like a King would mean something entirely different.

Before that, in the 16th and 17th centuries, Patagonia was almost a semi-mythical place. It was supposed to be inhabited by a race of giants called "Patagons". The area was so remote that this wasn't really proven wrong until the 18th or 19th centuries.

So depending on the time, it could mean he was living in a semi-mythical place, living as a leader of "savages", or living the rich life of a cattle baron. I suspect the author chose it just because it's a distinct sounding, far away place that we're all familiar with but know very little about. He might have said "Timbuktu" or "Tibet" or another such place just as easily.