Forget for a moment about "inferences" and consider only formulae.
Tautologies (or valid formulae) are formulae that are true in all interpretations, like A → A or ∀x(x=x).
If we consider one of Peano's axiom for arithmetic, like ∀x(x+0=x), it is not a "logical truth" (i.e. a valid formula) : it is true only in the "arithmetical" interpretation.
But the "logical laws", like ∀x(x=x), are still true in the "arithmetical" interpretation (they are ture in all interpreations !)
Of course, in the arithmetical interpretation also the theorem deriving from the arithmetical axioms will be true (they are logical consequences of the axioms).
Conclusion : adding axioms will reduce the number of possible interpretations, and vice versa.
The same for modal logics; we "specify" a normal modal systems adding some axiom to the "basic system K.
This amounts to adding "restrictive" requirements to the class of interpretations, and thus "reducing" the number of interpretations ...