Here’s where formal semantics really helps. I’ll assume we’re working in classical first-order logic with its standard model theory.
Assuming your axioms have a model, there is therefore an interpretation function in your model which among other things maps (or "assigns") each variable to a specific value. So let’s say the domain of your model is the natural numbers, and the interpretation function happens to map $x$ to the number $2$. So your open axiom really means $2 = 2$ in the semantics. So there’s no way to prove $3 = 3$ with that axiom. If you could, first-order logic would be unsound, which it isn’t, by the Soundness Theorem.
Here’s another way to go at it. Try using your axiom to prove $3 = 3$. You can’t apply your axiom because you can’t specialize $x$ to $3$ as you’d expect; only the universally quantified version of your axiom can do it: $\forall x(x = x)$.
So maybe you could write a theory with open axioms, but as you can see they aren't much use and don't capture what you're trying to capture.