How is "this" passed in C#

You've changed the Person instance that firstPerson.instance is pointing to, but not the original instance that firstPerson refers to.

So firstPerson is still pointing to the original Person instance (and so firstPerson.name returns the value set in the first instance), while firstPerson.instance is now pointing to a new (second) Person instance.

Person firstPerson = new Person();            // instance 1
Person secondPerson = firstPerson.myself;     // myself refers to instance 1

secondPerson.name = "Bill";                   // set name in instance 1
Console.WriteLine(firstPerson.name);          // get name from instance 1
Console.WriteLine(secondPerson.name);         // get name from myself in instance 1
Console.WriteLine(firstPerson.myself.name);   // get name from instance 1 (same as above)

firstPerson.myself = new Person();            // myself refers to instance 2, but firstPerson still refers to instance 1
Console.WriteLine(firstPerson.name);          // still getting name from instance 1
Console.WriteLine(secondPerson.name);         // still getting name from myself in instance 1
Console.WriteLine(firstPerson.myself.name);   // get name from instance 2 (since firstPerson.myself was reassigned)

firstPerson = new Person();                   // firstPerson and firstPerson.myself point to instance 3
Console.WriteLine(firstPerson.name);          // get name from instance 3, which is the default "Eddie"
Console.WriteLine(secondPerson.name);         // still points to instance 1, since that's what it was when it was assigned
Console.WriteLine(firstPerson.myself.name);   // get name from instance 3 (since firstPerson.myself is defaults to the new instance again)

this represent current instance of of a class.

When you are creating new instance of Person firstPerson.mySelf, that time it will refer to the new instance of Person class.

Person firstPerson = new Person();
Person secondPerson = firstPerson.myself; //Here you are referencing to same instance of Person class i.e. same `this`

But when you are creating new instance of Person, it will refer to new this

firstPerson.myself = new Person();  // New instance new `this`, but still `firstPerson` is referencing to previous instance

Explanation with diagram

In your case you created new instance of person and stored in myself property. but firstPerson and secondPerson is still pointing to same this instance

The myself is just a variable. So when you call

Person firstPerson = new Person();

you have 2 variables which point to the same instance: firstPerson and firstPerson.myself. With line

Person secondPerson = firstPerson.myself;

you introduce third variable which still points to the same instance. Now with

firstPerson.myself = new Person();

you create second instance and make firstPerson.myself point to this instance while the variables firstPerson and secondPerson still point to the first one.