Italians love baseball (¿!¿!?)

Yes.

You're correct that most immigrants who just stepped off the boat wouldn't even know what baseball was and would probably not "get" baseball, but baseball held sway back then in a way unimaginable today. Professional football and basketball hadn't yet caught on. It's the children of immigrants (like Tony) who would embrace baseball. (Unless I missed something when I stepped out for popcorn, Tony was the son of Italian immigrants and not an immigrant himself.)

In New York back in those days, there were three major league ball clubs: the New York Yankees, New York Giants, and Brooklyn Dodgers. My wife is Italian-American and her uncles (long deceased, sadly) were die-hard fans of the Brooklyn Dodgers and rooted intensely for right fielder Carl Furillo, a fellow Italian-American. The Yankees back then had Joe DiMaggio and Yogi Berra (among others) and the Giants had Sal Maglie (among others). Italians were discriminated against, so when an Italian-American ballplayer did well, it gave them pride and respectability.

Italian-Americans loved baseball during that period in American history (I'd say now as well though not quite as much). Baseball accomplishments were a great source of pride among Italian-Americans. There were many great MLB players of Italian descent - Joe & Dom DiMaggio, Yogi Berra, Tony Lazerri, Phil Rizzuto, Rocky Colavito, etc... - and it resulted in a lot of Italian-American fans, particularly in the New York area.

In BROOKLYN, Tony is portrayed as an enthusiastic Brooklyn Dodger fan. The Dodgers were a great team during the 1940s & 1950s and they were known for having perhaps the most loyal fans in all of baseball. The character, Tony, is right in line with that era.

Italians, in Brooklyn, from the 40s to the 50s... yes, they loved it. But in all fairness, all the greats were Italian back then.

It's an Italian-American stereotype

Young Italian men looking for heroes from their own culture, with whom they could identify, couldn't find them in too many places in America, but baseball was a great exception.