Shallow Copying Class Objects vs. String Variables

string a = "John";
string b = "Doe";

a = b; // Shallow copying strings
b = "Elon Musk";

Console.WriteLine(a); // Output: Doe

This prints "Doe", meaning the change in one variable is NOT reflected to the other variable. However, when "boxing in" the string property into a class and performing a shallow copy with class objects, the change in one object is actually reflected to the other object.

public class Person
{
    public string Name { get; set; }
}

// ----

Person person1 = new Person { Name = "John" };
Person person2 = person1; // Shallow copy

Console.WriteLine(person1.Name); // Output: John
Console.WriteLine(person2.Name); // Output: John

person2.Name = "Jane"; // Modifying the property via person2

Console.WriteLine(person1.Name); // Output: Jane (changes are reflected in person1)
Console.WriteLine(person2.Name); // Output: Jane

I am trying to understand what the reason is that the changes are reflected in 2nd case, but not reflected in 1st case. Both class objects and string type are reference-type, therefore, the change should be reflected in the 1st case too, right? Please help me understand what the difference between the two cases that is causing this difference in behavior.

I know that string type is immutable. Are class objects mutable? Is that the reason for the difference between two cases?

Note: I did check previously asked questions, and I could not find a similar question.