Well, my two parakeets smell like birds... which smells like.. feathers? Yeah, call me weird, but I've sniffed them. They smell like a down comforter. Chickens on the other hand, usually smell like dirt to me. Earthy instead of fluffy, but that's probably because chickens bathe in sand outside, while the parakeets like to bathe in the sink.
I clean their cage out weekly. The female of the pair I have now is picky about her perches. Her morning ritual is to jump down from the top perch and pick off every last piece of poo off the perches from the night before. Makes my job easier when it comes to cleaning off their toys. LOL
So I wouldn't say they stink, they just make the bird room smell like feathers.