I think around 80% it is due to a lifetime of hard work in a perfect environment (FC Barcelona's Academy) surrounded by the best people in the game. The other 20% may come from his psychological abilities, being able to reach very high levels of concentration at peak moments. Notice that he is not a very active and working player throughout the whole match, instead he shines through certain moments.
Lionel Messi was already known as an outlier as a young boy in Argentina. The psychological abilities and lifetime of training at Barca simply molded something that was already there, but there are hundreds of other footballers who trained at Barca the same time Messi did and none of them are Messi. Some of them are very very good, but none of them are Messi.
True, it's the 20% that makes him better than players like let's say Cesc Fabregas (same football academy) who is a top-level player but kind of lacks the magic. Also note that there is a thin layer between a very good player and a great one, sometimes measured in miliseconds and centimeters but the output is fantastic! :)
The interesting thing watching him play is that you attribute most of his game play to speed and balance due to his low center of gravity. I never thought he was that accurate because his motion when shooting doesn't seem very elegant.
In addition to the other reasons mentioned, I'd add that having a team of superstars built around you will help your stats no matter who you are. When your team has most of the possession and is consistently breaking down defenses, your stats will be high.
The article covers that. They compare Barcelona vs Spain. Both play the tiki-taka style and have mostly the same players. But Barcelona is significantly more efficient. The conclusion is that you need a player of his talents for that style to really pay off.
