@Theron125 ... interesting that you've never heard it. I have in both contexts: when talking about an enlarged heart as a medical condition and as a metaphor to indicate generosity. The most well-known example of the latter would be Dr. Seuss' "How the Grinch stole Christmas" where the story refers to the Grinch as having a heart three sizes too small.
That's for the thing that's owned. Remember, сердце is the subject of this sentence, and if this was a negative statement, it would be in genitive. You need to compare У моей сестры... with у меня… which is in genitive.
Does it help to compare the sentence with: "A big heart belongs to my sister"?
To build upon what chirelchirel said, what's possessed is going to be in the genitive in the presence of нет, but regardless of that the construction used to express possession is [У + genitive] which literally means "next to [...] is."
У меня есть карандаш. -> У + [я in the genitive] -> "next to me exists a pencil" or in other words "I have a pencil."
У собаки свитер. -> У + [собака in the genitive] -> "next to the dog is a sweater" or in other words "the dog has a sweater."