I'm not completely sure of that myself, but consider this analogy. Let PA+X be a formal system that consists of the axioms of PA plus a new axiom that introduces a new symbol X and simply says "X is an integer", without saying anything more about X. Then it's easy to prove "X is either even or odd" in PA+X, but it would be wrong to say that PA+X has a unique distinguished "standard model" that pins down the parity of X. So my statement about CH is more of a statement about our intuitions possibly misfiring when they say a formal system must have a unique standard model.