Second order logic, in first order set-theory: what gives?

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.

The predicate "is a real number" is absolute for transitive models of ZFC in the sense that if M and N are such models with M contained in N, then for every element x of M, the two models agree on whether x is a real number. But it can certainly happen than N has more real numbers than M; they just have to lie completely outside of M.

Example 1: If M is countable with respect to N, then obviously M doesn't contain all of N's reals.

Example 2 (perhaps more relevant to what you asked): Under mild large cardinal assumptions (existence of a measurable cardinal is sufficient), there exists a real number 0# (zero-sharp) which encodes the shortcomings of Gödel's Constructible Universe L. In particular 0# lies outside of L, so L does not contain all the reals.

Thus if you started with L and insisted on adding a measurable cardinal on top, you would have to also add more reals as well.