There are specific deduction systems that have only a finite number of logical axioms, or even no logical axioms. One example is natural deduction, which trades away logical axioms for a more complicated set of inference rules.
However, the use of infinite axiom schemes of logical axioms is not a problem for many of the applications of finite axiomatizability. If we have a structure $M$, or are constructing it, and we want to tell whether $M$ satisfies a theory $T$, there are two main issues we face:
- For each non-logical axiom of $T$, how do we tell that $M$ satisfies the axiom?
- If $T$ has infinitely many axioms, we have to quantify over them somehow.
For finitely axiomatizable theories, the second question is not an issue, so we can focus just on the first one. In any case, we don't need to worry whether our structure $M$ satisfies the logical axioms - we know it does, because they are true in every structure. We only need to worry about whether the structure satisfies the additional axioms of $T$. This is why finite axiomatizability is defined in terms of the non-logical axioms only.
If there are only finitely many new axioms in $T$, we can put all these axioms into a single sentence $\phi$ so that an arbitrary structure satisfies $T$ if and only if it satisfies $\phi$. That is a nice situation to be in, and it allows us in some cases to prove more about finitely axiomatizable theories than we would be able to prove about non-finitely-axiomatizable ones.
The existence of the single sentence $\phi$ also means there is a sort of "absoluteness" to whether a model satisfies $T$. If $T$ has an infinite axiom scheme, then from the point of view of a nonstandard model of arithmetic the scheme may well have nonstandard axioms - this is a subtle issue that can cause various kinds of confusion when people work with Gödel's incompleteness theorem. For a finitely axiomatized theory, because we can work with the single sentence $\phi$, even nonstandard models of arithmetic agree about the axioms for $T$.