I don't think he lacks emotional "toughness" as much as he lacks emotional maturity.
Unfortunately, he doesn't seem to have matured over the last four years. This is, however, not uncommon among overachievers in any walk of life and is observed in the business world all the time; once these people think or have others reinforce the notion that they are better than they are, they tend to lock in emotionally and developmentally at the point where the adulation began.
For Sanchez, this probably occurred sometime between the day when the Jets traded the farm to move up to Draft him and the first time Ryan called him the Sanchize, early in his Rookie season. The phenomenon was exacerbated by the fact that Pete Carroll, who knew something about coaching College students however badly he might suck as an NFL HC, told him that he should stay at USC for another year. So, Sanchez came to the league with a huge chip on his shoulder and his early NFL experience validated in his mind that he was right and Pete was wrong, leaving him closed to any other view.
Now, he's emotionally able only to blame just about everything and everyone but himself for his not being the next Peyton Brady.
Nothing has changed in my view of Sanchez from his first season. He is a genuinely qualified, "OK to Good" NFL QB, who, after the Jets let him go in a year or two, will have a long career in the League as a starter for small market teams that don't aspire to winning Super Bowls and eventually as a backup to an elite starter (though he won't be ready to accept that role for another six or seven years).
The tragedy is that Carroll was right and he should have waited before entering the League and he probably never should have come to a place that would feed and ultimately crush his ego to the degree that New York can do so. It is also unfortunate that he was coached at the outset by someone with emotional maturity "issues" himself, like Rex Ryan. In many ways, we'll never know what "might have been" had he followed a different path; but, that can be said of the careers of a lot of NFL players, particularly Quarterbacks.