The private box

The private box will be a type that implements the abstract base class. The goal of the private box is to provide a wrapper around the concrete object implementing the PAT and forward all of the calls to the boxed object.

Following the conventions also suggests that you should use _Any#MY_PROTOCOL#Box as a class name for this component:

private final class _AnyAnimalBox<A: Animal>: _AnyAnimalBase<A.FoodType> {
    // The target object, that is an Animal
    var target: A
    init(_ target: A) {
        self.target = target
    }
    
    // Overrides of the abstract classes's implementation
    // Forward all invocations to the concrete target
    override var name: String {
        return target.name
    }

    override var preferredFood: A.FoodType? {
        get { return target.preferredFood }
        set { target.preferredFood = newValue }
    }
    
    override func eat(food: A.FoodType) {
        target.eat(food: food)
    }
}

As the _AnyAnimalBox class extends _AnyAnimalBase, we will be able to keep references of our boxes as their abstract class type, which successfully bridges the world of protocols and their associated types to the world of generics.