'Baby daddy' is a new slang expression in some parts of the U.S. It's used when a woman has children by different men. John may be the baby daddy of her oldest, David may be the baby daddy of her daughter, and the father of her baby is her her live-in boyfriend. It does not have quite the same meaning as 'father' because the baby daddy does not live with his child and generally does not have much interaction with the child. it's not an expression I would use for language learning.