It's an interesting question, in spirit at least, although you could make it more precise by saying exactly which algebraic-style axioms you're talking about---there are several proposals.

Here's something between a comment and an answer. To avoid the question of which axiom system we're using, I'll just talk about 2-categories (weak ones, i.e. bicategories). 2-categories are defined in such a way that "all diagrams commute", and the same goes for n-categories in general. Sure, there's a finite axiomatization, but we only know it's the right one because it allows us to prove that "all" diagrams commute.

You can write a computer program that spits out, in turn, all the diagrams that are supposed to commute. In that sense, the axioms are recursively enumerable.

An interesting observation is that 2-categories can be finitely axiomatized. In fact, this is the axiomatization that everyone meets: you have a pentagon, and a triangle, and some naturality squares. But in principle you have an infinite collection of coherence axioms: the "all diagrams" I referred to.

So the theory of 2-categories is finitely axiomatizable, but I don't know of any explanation of why it had to be. More generally, if you take a finitely axiomatizable algebraic theory (e.g. monoids) and categorify it (obtaining e.g. monoidal categories), I don't know whether that categorified theory must inevitably be finitely axiomatizable.

(Of course, finite axiomatizability is not the same as computability, but it's a closely related question.)

It's an interesting question, although you could make it more precise by saying exactly which algebraic-style axioms you're talking about---there are several proposals.

Here's something between a comment and an answer. To avoid the question of which axiom system we're using, I'll just talk about 2-categories (weak ones, i.e. bicategories). 2-categories are defined in such a way that "all diagrams commute", and the same goes for n-categories in general. Sure, there's a finite axiomatization, but we only know it's the right one because it allows us to prove that "all" diagrams commute.

You can write a computer program that spits out, in turn, all the diagrams that are supposed to commute. In that sense, the axioms are recursively enumerable.

An interesting observation is that 2-categories can be finitely axiomatized. In fact, this is the axiomatization that everyone meets: you have a pentagon, and a triangle, and some naturality squares. But in principle you have an infinite collection of coherence axioms: the "all diagrams" I referred to.

So the theory of 2-categories is finitely axiomatizable, but I don't know of any explanation of why it had to be. More generally, if you take a finitely axiomatizable algebraic theory (e.g. monoids) and categorify it (obtaining e.g. monoidal categories), I don't know whether that categorified theory must inevitably be finitely axiomatizable.

(Of course, finite axiomatizability is not the same as computability, but it's a closely related question.)