If we consider a single schema as a single axiom, so ZFC for example would be finitely axiomatizable after this kind of counting axioms.
By schema its meant a syntactical expression (string of symbols) that contains among its symbols free meta-theoretic variable symbols that range over a decidable set of formulas of the language of the theory in such a manner that each substitution of all those metatheoretic variables in the expression by instances (formulas) from what they range over would result in a sentence of the language of the theory. For example the separation schema of Zermelo written as: $$\forall A \exists x \forall y (y \in x \leftrightarrow y \in A \land \phi) \text { is an axiom }$$; Where $\phi$ is a metatheoretic variable that ranges over all formulas of the language in which $A$ doesn't occur free.
This is considered as a single axiom. While the separation schema written as: $$for \ n=1,2,3,...\\ \forall p_1,.., \forall p_n \forall A \exists x \forall y (y \in x \leftrightarrow y \in A \land \phi)$$, that is not a single schema, it is an infinite collection of schemas, so it is not considered as a single axiom.
Now is it the case that every effectively generated first order theory (with finitely many primitives) is finitely axiomatizable in this sense?