New York City might have more birds than people (and, well, let's not talk about the rats or the bedbugs, okay?) so it's only natural that they show up a few times in a story set in New York City.
The most notable instance of birds of a feather flocking together is when Oskar turns on Mr. Black's hearing aid and "A flock of birds flew by the window, extremely fast and incredibly close" (7.126). These birds seem to punctuate this fateful encounter between a nine-year-old boy and a man who's well over a hundred. Their relationship ends up changing both their lives. We even get treated to a picture of a flock of birds smack dab in the middle of that scene.
Birds are important to Oskar because of one of his more creative inventions: the birdseed shirt. We doubt Oskar would have invented this were he not constantly picturing his Dad falling to his death from the World Trade Center. The birdseed shirt is the ultimate in safety (if not fashion): birds will latch on to the wearer's shirt and fly him or her out of harm's way.
In order for the plan to work, however, you need birds. Lots of them. Oskar has a plan to keep all the birds in New York City from hitting buildings, but there's one problem: "The birds would never leave Manhattan" (13.103). Oskar says that would be great, though. For him, safety is more important than exploration and adventure. Birds, which are often a symbol of freedom, suddenly become a symbol of captivity in this example.