Reflexivity axiom for inferring functional dependencies
"What does it really mean that some attribute is a subset of some other attribute?"

It means nothing.  Functional dependencies exist between SETS of attributes.

Whenever the treatment is supposed to be formal and precise (as in, e.g., when stating the very definition of the axioms), that's how X,Y and Z are supposed to be interpreted.

Unfortunately, people tend to always write down concrete examples of FD's without making the SET property explicit.  As in, e.g., AB->C, where A,B and C are attributes.  Formally, that should actually be {A B} -> {C}, but those braces don't add very much.

And your formulation of the augmentation axiom is incorrect precisely on that count.  If X,Y and Z are sets of attributes, then the only correct formulation of the augmentation axiom is

X -> Y   ===>   (X UNION Z) -> (Y UNION Z)

The reflexivity axiom provides a formal definition of what it means to be a trivial FD.  The various normal forms are defined in terms of nontrivial FDs, hence it must be pinned down what that means exactly.  The reflexivity axiom does that.