You'd do it just like your original post:
Stream &pc = Serial;
Stream &gsm = Serial1;
That's perfectly valid. But, Serial_ inherits directly from Steam. It apparently does not require a .begin() method. And, you have been told Stream does not have a .begin() method. You can't do what you're trying to do because what you're trying to do want is simply wrong.
The answer was given to you in Reply #8. The setup() code should call the .begin() -- if required -- on the specific object (not on a reference to it, not on a base class) right at start up. I'd probably use pointers to Stream rather than references. That way you could assign them in the same part of setup(). After that, the remainder of the code is agnostic to the actual type of Stream being pointed to.
If you want to support multiple boards in the same code, use compiler '#if' directives to provide the required versions of the above setup() functionality.