Why would the woman give her daughter the swan?

It is a parable "Feathers from a Thousand Li Away”. You can check out the story below. The swan is a symbol of "a creature that became more than what was hoped for. " The woman gives her daughter a swan because in America, she will grow to be more beautiful and successful than ever thought possible. Her daughter will not rely on a husband but transform into her own beautiful person,

"In America I will have a daughter just like me. But over there nobody will say her worth is measured by the loudness of her husband's belch. Over there nobody will look down on her, because I will make her speak only perfect American English. And over there she will always be too full to swallow any sorrow!"

You can check out the whole story at the link below,

http://nelli-ganenko.narod.ru/000004_0.html