The Red Sox "Stole" Sweet Caroline From A Canadian Beer Ad!
Beautiful Girls came out in 1996.
Wedding DJ's conducted Bum Bum Bum sing-a-longs as early as 1997.
This beer commercial is from 2001.
The Red Sox started playing the song nightly in 2002.
The Mets, as do the Rangers, have every right to play it, for the simple reason that it's fun, as pointed out by Fafif's Greg in the comments section.