Is my code wrong or is SoftwareSerial not so reliable?
I don't see this as an either/or question. The code has some issues, but they don't relate to serial, and SoftwareSerial is quite reliable.
What is not defined is what is connected to what pins, and what the problem is.
AnotherClass aClass;
The aClass variable is an object.
aClass = new AnotherClass(&softSerial); // This will get values on update, but not the right values (pin 2&3).
aClass = new AnotherClass(NULL); // This will get the right values on update (using rx&tx).
The new operator returns a pointer to an instance, not an object.
AnotherClass soft(&softSerial);
AnotherClass hard(NULL);
is how the instances should be created.