Yeah, but if he was named "the blue stingray" he'd be blue and fishy...
It's not necessarily the "costume" that does all the defining... but each character has to be defined in some way like that, you know? If there's not something about themselves that stands out enough to account for the crazy name then it has to be the costume etc.
That's how things work.
That reminds me of something from a comic book actually: there's the Iron Man villain, Iron Monger (who you may remember as "The Dude"/main villain from the Iron Man film). He was called that because, well, he sold Iron for a living, and decided to make a suit out of it and do crimes. Years later he's replaced by this kid who's a bit of a tech whiz and also a bit crazy. Marvel pretty much just wanted to keep the name. However, it didn't make any sense, since "Monger" isn't an inherently dangerous or evil term: it just sounds kind of nasty. They eventually started poking fun at it when the kid appeared as the villain in a Spider-Man comic. The name of a character is important, I think. If you're called a certain thing, then what you do or how you dress etc. should reflect that in some way. Kind of like how a person gets a nickname: it's usually based on something they've done or a way they look. Unless it's an ironic nickname, but whatever.
I don't really feel too strongly either way on this particular issue. I've nothing against the idea of Harley Quinn dressing up in a nurse's outfit while wandering around a mental health facility: it's within her character to do so, or within the Joker's to have her do it. I just don't think the design itself is very good.