When we talk about an axiom, shouldn't it be a group of axioms since we have an axiom for each variable?.
For example, "B ⇒ (C ⇒B)" is an axiom schema standing for an infinite number of axioms (Introduction to Mathematical Logic, Mendelson). Therefore, x1 * 0=0 should be an axiom schema too, since it stands for an infinite number of variables and therefore we have an infinite number of axioms. Am I wrong?
$x * 0 = 0$ is really shorthand for the claim $\forall x \ x * 0 = 0$. So this is really just one specific first-order logic claim, and therefore one specific axiom, not an axiom schema like $B \to (A \to B)$.
You will often see Greek letters being used as statement/formula variables used in mathematically laying out axioms. So since different FOL formulas can be filled in for those variable we are really dealing with an axiom schema. So you'll often see something like $\varphi \to (\psi\to \varphi)$, and now you immediately know you're dealing with an axiom schema.
But note that we don't have anything like $\varphi * 0 = 0$. Instead we have $x * 0 = 0$, and $x$ here is not a statement/formula variable, but an object variable as part of the very language of FOL. Sure, it has many instances when applying the universal elimination rule, but that is not what makes it a schema. It is still one specific expression in the language of FOL.
Indeed, all of the first 6 axioms of Peabno Arithmetic (I am guessing that is what you are looking at) are specific FOL statements, and therefore specific axioms, not axiom schemas. The induction axiom, however, is an axiom schema :
$(\varphi(0) \land \forall x (\varphi(x) \to \varphi(s(x)))) \to \forall x \ \varphi(x)$
Here, the $\varphi(x)$ is any formula that has $x$ as a free variable, and so this axiom schema has an infinite number of instances which are genuine FOL statements.
Of course, if we are doing second-order logic, we can write:
$\forall \varphi ((\varphi(0) \land \forall x (\varphi(x) \to \varphi(s(x)))) \to \forall x \ \varphi(x))$
and that is one specific SOL axiom, and therefore not an axiom schema for SOL.