In the first movie, Falcone said he was the prince of Gotham and would have to go over 1000 miles (Italy being well over 1000 miles away) for anyone not to recognize him, so that part was mostly a joke..
Catwoman was looking for the "Fresh Start" program or whatever it was which would have allowed her to create a whole new identity, free from the record she had from having had to steal and commit other crimes. Batman said he'd give it to her if she helped him, and since she's there with him at the end, I'm assuming they both used it to create completely believable new identities for themselves. If anyone ever came up to him and said he looked like Bruce Wayne, he'd just play it off, saying he had heard that for years. We know Wayne is great at pretending to be people since he 'pretended' to be Bruce Wayne for all those years -- always putting on the show of the disconnected billionaire while never revealing who he was.