No, it doesn't mean that. The interface doesn't actually contain the property, either.
Remember than interface has to be implemented by a class in order for you to use it. An interface only defines a contract, meaning that any class that implements that interface will have all of the members defined by the interface. The interface doesn't actually have those members; it's more like a template. The instance of the class that implements the interface is the one that contains the property, and therefore the private backing field used for the property.
Interfaces can only define signatures for the following members:
Methods
Properties
Indexers
Events