How to determine method caller type at run time?

I wish to know if there is a way for a method to determine, at run time, the type that invokes it.

[UPD2] For this particular case I'm interested in the Type of the caller only, not the name of the method, so, other similar questions like How can I find the method that called the current method? or Retrieving the calling method name from within a method doesn't solve my problem.

For the sake of simplicity, let's assume I have a class named AnimalContainer. At its initialization its constructor will receive a list of allowed types: The types who are allowed to register animals to the container.

The goal of this class (something like an IoC Container) is to serve as a container for any type who wish to retrieve an instance of a particular animal. The restriction is that only an arbitrary set of types should be able to register animals to the container. Then, if a non-allowed type calls the method, the method would return an InvalidOperationException

Expecting a Type or an object parameter:

void RegisterAnimal(Type callerType, Animal animal) {...}
// Or
void RegisterAnimal(object invoker, Animal animal) {...}

These won't work because the invoker is not restricted to specify its own type (or instance). For example:

// Inside class A:
animalContainer.RegisterAnimal(typeof(B), anyAnimal);

[UPD1] I've seen many answers stating that this seems so be a bad approach. If you consider so, would you suggest another way? In general terms, What I'm trying to achieve is to code defensively. I surely can have, by convention, the set of Types that are allowed to register animals in that container. What would happen, is any other developer, who aren't aware of this convetions, try to register an animal inside a class doesn't suppose to be doing that? It will be doing something that the design isn't expecting, so, most probably will end up by introducing some erros. This is just the simpler example I could find (The AnimalContainer) but it can be extended to a huge set of situations:

1. Having a set of controllers clases that would register dependencies into an IoC Container.
2. Having a MessageContainer that register publishers and subscriptor for messages between different types (decouple the fact that an event subscriber should know who is publishing the event). And then specifying that only a set of types can publish a given message. Ex: Any class can listen for NewAnimalArrive message, but only a specific set of clases can fire that event.