Continuity mistake: Just before Batman takes his mask off at the end, the black makeup around his eyes vanishes while the mask is still on. (01:48:20)
Plot hole: When The Penguin is controlling the Batmobile, Batman punches through the floor to take off the transmitter. We still see a video feed of The Penguin. Why? Batman pulled off the transmitter, so there is no reason for there to still be a video feed.
Visible crew/equipment: After Batman and Catwoman's mistletoe banter, a wire can be seen attached to his back as he glides off the rooftop. (01:20:15)
Continuity mistake: In Schreck's department store, when Catwoman is doing her cartwheels, her heels collapse. (00:53:40)
Revealing mistake: In one scene, the Penguin is making a "bat" shadow puppet on the ceiling. But there's no light behind his hands. If there's no light, there's no shadow. (01:03:45)
Deliberate mistake: When the penguin goes into the cemetery, he bumps a tombstone and it moves. Almost certainly a tribute to legendarily poor director Ed Wood, who Tim Burton later directed a biopic of - this exact thing happened in one of his films. (00:39:15)
Continuity mistake: When the Penguin is looking out of the barred sewer hole, from an inside shot we see the penguin holding two bars, and in between the bars he is holding are two bars. In the next shot, there are four bars in between. (00:06:05)
Continuity mistake: In Batman's first battle with the Red Triangle Circus gang he manages to escape from a fire eater by turning his batmobile around and burning the guy with the rear exhaust jet. He does this by flicking a switch (outlined in caution symbols) next to the steering wheel. When the Penguin takes Batman for a ride there is a television screen where the switch is supposed to be. When Batman blows the Batmobile apart to get through a narrow alley the switch is back in place. Not only that but the switch seems to control just about every major system on the car including the lights, shield cover, gun turrets (in the first movie), canopy etc.
Continuity mistake: When Selina puts her soft toys into the sink, and begins to push them down into the shredder hole, between shots the positions of the teddies in the sink differ. (00:29:45)
Continuity mistake: Catwoman's lipstick smears down onto her chin when she puts The Penguin's bird in her mouth, yet in the shot just before she gives herself a "bath", the lipstick momentarily is gone. (01:03:20)
Continuity mistake: When the Penguin bites the guy on the nose, while he is talking to Max Shreck afterward the blood on his face is there, then it disappears, then it's there again. (00:49:40)
Continuity mistake: When Selina retrieves her papers from the floor, she places them in a disorderly pile beside the clown. In the shot when she zaps the clown with the taser, the papers have changed position and are stacked neatly. (00:15:55)
Revealing mistake: When the Penguin drives the Batmobile remotely, the cars it smashes are damaged just before it hits them. (01:20:50)
Continuity mistake: When Selina returns home from work she takes milk out of the fridge and pours her cat some, leaving the carton on the worktop. She then runs to work, and when she returns after being changed to 'catwoman', the milk has vanished from the worktop. The dishes also move and plant pots appear on the window sill. (00:22:50 - 00:28:45)
Continuity mistake: When Selina becomes enraged and trashes her apartment, she grabs a frying pan off of the oven hob, which was not there previously, when she entered her apartment. (00:28:40)
Revealing mistake: When The Penguin shoots the clown who says "Killing sleeping children...", as the clown falls back to the floor, you can clearly see he falls on a wrinkled, soft mat which has unsuccessfully tried to be disguised. (01:29:30)
Visible crew/equipment: When the large present drives out into the middle of the Christmas procession, as the lid of the gift springs up, a long white wire can be see attached to the lid, which wasn't there beforehand. (00:11:30)
Revealing mistake: The snow that they use floats on water in almost every scene. (00:02:00)
Audio problem: The Batmobile is supposed to be powered by a Jet engine. The sound in the sequel is dubbed rather flawlessly except for two brief instances when you can hear the 8 cylinder piston engine.
Revealing mistake: When Batman glides off the rooftop we catch a glimpse of his face as he flies over Gotham. It isn't Michael Keaton, it's a rather crude looking dummy. (01:20:15)
Answer: The film keeps it ambiguous. There are two ways to look at it. Scenario A- She literally died and is brought back to life by the alley-cats somehow, adapting some of their traits. Or... Scenario B- The trauma of falling and sustaining a head-injury, along with her paranoia after Schreck tried to kill her, has driven her mad, and she uses her connection with cats to build a new persona. (Which is supported by the fact that all the things she claims "kills" her with each of her "nine lives" wouldn't actually have killed her. Ex. Her one fall is broken by the kitty-litter truck. Max doesn't hit her in any vital organs when he shoots her. Etc.) Her destroying her apartment is her lashing out at all the things she used to hold dear- her vision of a normal life, etc. It's symbolic of her purging the past and embracing the future. (Plus, oftentimes when people throw fits, they'll smash stuff up).