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 phoenix is also known as a symbol of royalty. Phoenix Symbol-Meaning The phoenix or 'fire bird' is a mythical bird and is a symbol of the sun.Other mythical birds who resemble the Phoenix and appear in other cultures include: Persian Simurgh (also known as Angha). Chinese Fenghuang (also known as the Chinese Rooster or August Rooster).Russian (Slavic) Firebird.he bird is generally thought to be a symbol of freedom. They can walk on the earth and swim in the sea as humans do but they also have the ability to soar into the sky. Birds are free to roam to earth and the sky. Many cultures believe that they are a symbol of eternal life; the link between heaven and earth.