Paul McCartney and the rest of the Beatles most certainly grew up hearing Eurasian Blackbirds. Their song is beautiful, so it's no wonder the Beatles chose to weave it into one of their songs.
But McCartney wasn't singing about the bird. He was singing about the racial strife in the American South in the 1960s.
As he said later, "This was really a song from me to a black woman, experiencing these problems in the States: 'keep your faith; there is hope.'" Does the Eurasian Blackbird really sing in the dead of night? Generally not. Still ... what a beautiful, hopeful song.