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.
Youâve Come A Long Way, Baby. Belly dancing may have originated as a fertility ritual. 17,000 year-old rock engravings found in the caves of Addaura, near Palermo in Sicily, depict what appears to be a ritual fertility dance, as do ancient Egyptian tomb paintings and Greek sculptures. In Sparta, women danced for Artemis, who was a goddess of the moon and of fertility.