Oh, yes. they are slang expressions, certainly - but "having a bird," "having a cow," "flipping out," "blowing [one's] top" all basically mean the same thing in my part of the world. I wouldn't define them as "is crazy" in the literal sense of "Is mentally unstable and should be institutionalized," however, but rather as "losing control" or "becoming extremely enraged/visibly frustrated" or something like that. [Native US English speaker here.]
Hi, I just wanted to confirm that it indeed means "someone is crazy". The idea is that you have a bird sitting in your head that interferes with the functioning of your brain :) Pons gives "to have a screw loose" as a translation. The expression "eine Schraube locker haben" is also used in German in the same sense.
@Soglio: It isn't necessarily a permanent state. It's something you say when people do things you consider somewhat crazy. Anger or an emotional eruption is normally not implied. "Eine Schraube locker haben" and "Einen Vogel haben" are synonymes in German and seem to fit well with "to have a screw loose". (At least that's what I get from Pons. The existence of this phrase in English is something I just learned, so I dont' have much experience with using it, yet. But I'll try it out on a bunch of people and report back :) )