In Funny Girl, why does Fannie call herself Sadie married lady?

There is the song "Sadie Sadie" in the movie/musical Funny Girl with Barbara Streisand. The main character is Fannie Brice, but she sings 'I'm Sadie Sadie married lady'. It's a really cute song and I love it - but it just doesn't make sense to me she's calling herself by a different name?