Gustavo Fring is not Chilean, but he may have lived in Chile. He probably did live in Chile, but definetely wasn't born here as he claims.
I'm Chilean and I've never even seen a coatí (or eaten a lucuma) and when he mantioned the animal in the most recent episode my wife and I almost at the same time said "he's not from here".
Plus, he's black. Blacks abound now in Chile, but before the 90s or even the 00s blacks were very rare. Rare like coatí. Blacks are such a new addition to the Chilean population that if you see a black person here and you assume that person is an immigrant, you're right at least 90% of the time.
The tree he spoke about is also much more abundant in other places, although I know they exist here.
But he did make paila marina (twice) in Breaking Bad and that's a very Chilean dish. As he said it's just a fancy name for fish stew but the ingredients, temperature while cooking, temperature while serving and and timings are very precise, and no one that's not Chilean or knows about Chile would make such a random, obscure regional dish or call it by that name.
That, and also his friend Maximino Arciniega was Chilean and studied in USACH (that's Universidad de Santiago de Chile, it's a real U), and the actor that played Max even studied the Chilean accent. You can tell, because he made the "s" and "d" sounds at the end of words almost silent, and although his accent wasn't perfect it was really unmistakenly Chilean (good actor, he could have gotten away without the extra effort of doing an accent). Gustavo's Spanish is very broken and most times I find myself reading the English subtitles, but they may explain that later. Maybe he forgot most of it.
What I think is this: he was born in some other country (where lucuma trees, coatí and black people were more common than in Chile in the 70s and 80s) , used his genius and charm and resolve to escape poverty but did it like a smooth predator, and because he has no morals he became, by less than legal means, very powerful very young and was able to fly under the radar of the law. But someone didn't like that, he must have earned enemies, and after a while he needed to fly somewhere else and get a new identity. The dictatorship of Pinochet offered him an excuse to exist without much of a past when he moves to America. If you're a bad guy it's smart to claim a well known and badder guy was opressing you.