and i know a brother and sister that sang "anything you can do (i can do better" from annie get your gun. it really works that way actually.
EDIT-
also, from r&h's cinderlla.. i don't know what the actual title is, but the song the 2 stepsisters sing.. "why would a fellow want a girl like her, a girl who's merely lovely, why can't a fellow even once prefer, a girl who's merely me.. what's the matter with the men.."
that is cute. not rivalry, but.. siblings.
hear my song; it was made for the time when you don't know where to go, listen to the song that i sing, you'll be fine..
There's a song by Irving Berlin called "Sisters," it's a great little number and there is a recording floating around with Rosemary Clooney and Doris Day singing it.
I'm not sure if Rosemary ever recorded "Sisters" with Doris. The most popular version is the one she did with her own sister. There's always "Sister Kate," but that might be more of an, "I envy my sibling" song.
It's like writing "Norma Shearer for the win!" in a Joan Crawford biography.
"Anything You Can Do" would work great "Sisters"...sure, why not? "What is This Feeling" might even work. It could be interesting coming from siblings.