from the origin of "rose" or "rosie," Rosee is the perfect name to name a pet, especially if the owners name ends with two e's, like "Destinee." It shows you are a good owner of your beloved pet if you name them similar to your name.
While walking down the street...
Stranger: Oh what's your dogs name?
Destinee: It's Rosee! ...with two e's
Stranger: Wow you must be a good owner!
Rosee: WOOF WOOF!