Interesting. I didn't know the words origins. I've always considered it alongside the classic English tradition of referring to women as various animals.
This isnt a uniquely female experience. Men commonly refer to their friends as "dog". Also, in the gay community, men's body types are equated to animals such as "bear," "otter," and "cub".
As someone not too familiar with American cultures, I'd probably make an assumption and go for the (to me) more masculine bird over the docile and flower loving bee, since bees have stingers that they normally would never use and birds have beaks/peckers.
Hmm, well, I have heard women being compared to singing birds (or more degrading as vultures or pen of hens if in group), but I've more often heard women being romantically compared to bees or flowers. Though, I don't think I've ever heard men being compared to bees, but often to birds (eagles, vultures, seagulls, etc.).
Might also be local culture, as I usually think of harmony, nature, and perhaps matriarchy when pondering bees, while birds seem much more gender neutral, like, standoff-ish, elegant, brutal, impulsive, egoistic, even presented as predatory and evil in children movies and some media.
So, using common stereotyping, you can see where I'm coming from.
I've heard dame used more often than bird myself. Honestly, not sure I've actually heard bird used... it's like a vague sense of "I think I knew that.... right?" and my brain shrugs back.
It's slang you'd hear 50 years ago in the east end and Essex. You'll only really hear it used by gangsters in movies these days or someone putting on the accent for laughs, possibly from an old geezer, you certainly won't hear it used by respectable establishments or family friendly media. It's not generally considered offensive but is considered uncouth.