Whether they have any extant function or not, they have lost their former function and in that sense they do fit the definition of vestigiality. Vestigial structures are often called vestigial organs, although many of them are not actually organs.
There are quite a few birds famous for doing ritualized mating dances, so I would only list 5: The Red Capped Manakin, The Black-footed albatross, The Western and Clark's grebe, The Superb bird of Paradise, & The Victoria's riflebird.
The phrase is evocative of the metaphors and euphemisms often used to avoid speaking openly and technically about the subject. According to tradition, the birds and the bees is a metaphorical story sometimes told to children in an attempt to explain the mechanics and consequence of sexual intercourse.