That's how it works already. Strings are a reference type- your variable A is a reference (like a pointer) to a string on the heap, and you are just copying the pointer's value (the address of the string) into the variable B.
Your example doesn't change the value of A when you assign "abcd" to B because strings are treated specially in .net. They are immutable, as Kevin points out- but it is also important to note that they have value type semantics, that is assignments always result in the reference pointing to a new string, and doesn't change the value of the existing string stored in the variable.
If, instead of Strings, you used (for example) cars, and changed a property, you'd see this is the case:
public class Car {
public String Color { get; set; }
}
Car A = new Car { Color = "Red" };
Car B = A;
B.Color = "Blue";
Console.WriteLine(A.Color); // Prints "Blue"
// What you are doing with the strings in your example is the equivalent of:
Car C = A;
C = new Car { Color = "Black" };
It's probably worth noting that it does not work this way for value types (integers, doubles, floats, longs, decimals, booleans, structs, etc). Those are copied by value, unless they are boxed as an Object
.