I suspect that if it were a player on an opposing team (let's say, a member of the Red Sox, the main Yankee rival) who notched a milestone hit at Yankee Stadium and the same fellow caught the ball it in the bleachers, the guy would heartily laugh at the mere suggestion that he give the ball away. ....especially if he knew its worth (about a half million dollars on the collector market). That thing would be on eBay before 7th inning stretch. Yes, that's speculation on my part, but one based on decades of observing people who idolize certain athletes. And the guy in question has admitted to being a HUGE Jeter fan(atic).
Those who think he'd give the ball away to a David Ortiz or a Kevin Youkilis or some other hated Red Sox player are laughably naive. His offering to Jeter wasn't "class" -- it's sports idol worship taken to a ridiculous extreme.
Ding, ding, ding! We have a winner!