I still don't understand "why" the middle finger is called the bird in English... Oh, wait. Maybe it's a direct translation of an Italian expression? After all, “Between 1900 and 1915, 3 million Italians immigrated to America”
In Italian, the colloquial, and rude word for penis is "uccello" which means bird.
From Late Latin aucellus, contracted form of Vulgar Latin *avicellus, diminutive of Latin avis, from Proto-Italic *awis, from Proto-Indo-European *h₂éwis. Compare French oiseau, Catalan ocell, Occitan aucèl, Neapolitan auciello, Sicilian aceddu, Friulian uciel, Venetian oxeło, Ligurian öxéllo, Romansch utschè.
In some Spanish dialects, a penis is commonly called a pájaro (bird), and in Catalan, the term polla (young chicken) is used.
The slang meaning of bird is also used in several Asian languages
A calque from various Asian languages that use "bird" as a slang term for "penis", including Malay burung, Chinese 鳥, and Tagalog ibon.
- (Asian slang) A penis.
Of significant relevance is the English and vulgar term for penis, cock, the "male of the domestic fowl" and not originally the shortened form of cockerel. Since the 1610s, the term cock has referred to the penis in the English language.
The male of the domestic fowl (along with the bull) has been associated in many lands since ancient times with male vigor and especially the membrum virile, but the exact connection is not clear (the cock actually has no penis) unless it be his role as fertilizer of the domestic hens, and there may be some influence from cock (n.2) in the "tap" sense.
Many thanks to @user10518 who posted the following excerpt in the comments below.