Notice the thug says what are you, not who are you? Batman is so mysterious at this point, that people aren't even sure if he is human.
VICKI: Could you tell me which of these guys is Bruce Wayne?
BRUCE: I'm not sure.
Bruce's response is a flirty joke, playing around with Vicki Vale so he can later surprise her that he is, in fact, Bruce Wayne. But if you read into this even more (which we love to do) you can see a little bit of Bruce's existential crisis coming out. If he isn't sure whether to be Bruce or Batman, maybe he isn't Bruce Wayne anymore.
KNOX: The guy's a stiff. He's a rich stiff, but you could do much better.
Knox wants to win Vicki over by putting down Bruce. It doesn't work. Later in the movie, Knox, with a baseball bat, fights off a bunch of bad guys. We're not sure if he does it out of the goodness of his heart, to impress Vicki, or a little of both, but it shows a bit of character evolution in his identity. He goes from a self-serving journalist hack, to a self-serving journalist hack who's willing to risk his life to fight crime. He and Bruce have something in common.
JOKER: I've been dead once already. It's very liberating. […] Jack is dead, my friend. You can call me... Joker. And as you can see... I'm a lot happier.
After death, Jack Napier is reborn as Joker. He says he's a lot happier, and that's true. After he is dead, he has very little cares in the world. He has the freedom to do whatever he wants, no matter the cost to society. That would make any homicidal maniac happy.
JOKER: I am the world's first fully-functioning homicidal artist.
Um, good for you? Gold star?
THUG: He's human after all.
It is a shock to the criminal element Gotham to learn that Batman is human. Do you think that makes him more or less scary to them, to know that he's a guy in a bat suit instead of a man-shaped bat?
VICKI: A lot of people think you're as dangerous as Joker. […] You're not exactly normal. Are you?
BATMAN: It's not exactly a normal world, is it?
Vicki makes a good point here, calling out the parallel between Joker and Batman in terms of craziness. And Batman makes a good counterpoint. In an abnormal world, it takes a crazy hero to fight a crazy villain.
BRUCE: You know how people have different sides to their personality? Sometimes... a person will have to actually lead a different life.
Bruce attempts to tell Vicki that he is leading a double life. One is an eccentric billionaire. The other is an eccentric bat-suited crime fighter. Bruce acts like these are two different sides to their personality, but might there be some overlap between the two?
JOKER: But one thing I am not is a killer! I'm an artist.
Joker doesn't want to be seen as a common killer, so he attempts to elevate himself to the status of artist. We can also say that Joker doesn't want to be seen as common anything, so everything he does, he does big. You have to give the man credit for going all out on everything from his wardrobe, to his choice in women, to his crimes to murder an entire city. Go big or go home.