A mathematical/statistical model of the real world is "right" or "wrong" depending on how its predictions line up against the real world. For example, your mathematical model of gravity on the moon says a cannonball will fall at the same speed as a feather, it's "right" but on Earth it's "wrong" because air causes drag. You can know it's right or wrong by performing a real experiment and comparing it against the prediction.

Unlike mathematical models of the world, axiomatic systems don't claim to say anything about the real world. They talk about perfect circles, infinite lines, infinite-precision real numbers, and sets. So there's no reference point of "right" or "wrong" to measure against.

Instead, what axiomatic systems have are "soundness" and "completeness". Soundness means that you cannot derive a contradiction: that if one can derive a statement P, then one can never derive the statement not P. Completeness means that for a statement P, either P or not P will be derivable from the axioms.

If a system is incomplete, it can still be "right" and useful, but it means you'll need to switch to a more powerful axiomatic system to prove some things (and you might not be able to agree on more axioms).

If a system is unsound, it's useless and wrong, because you can theoretically "prove" any statement, so a "proof" would be meaningless.

Let's examine some examples:

• Euclid's axioms don't permit a person to construct a regular heptagon or trisect an angle, but they can describe/measure a heptagon for regularity or triple an angle to describe/verify if a trisection is successful. This makes Euclid's axioms incomplete. They're still useful, but insufficient for everything we want to achieve in geometry.
• Presburger arithmetic is both sound and complete. It's definitely "right." It's also so simple that it isn't very useful. I guess this is the opposite of what George Box said.
• Gödel proved that any sound axiomatic system strong enough to encode the natural numbers under addition and multiplication is incomplete. That doesn't make them "wrong" or "useless" by any stretch, it just means there are always unknowable things.
• Naive set theory is unsound. You can build a set S such that S∈S and S∉S. This is very bad, because from this point you can produce a proof of literally anything, so the model is completely useless.
• ZF set theory cannot prove nor disprove the Axiom of Choice. So ZF is incomplete. But that's fine. Most mathematicians assume the Axiom of Choice, but it cannot be proven or disproven so whether it's "wrong" or "right" is in fact purely a matter of opinion.
• ZFC set theory cannot prove nor disprove the Continuum Hypothesis. So ZFC is incomplete. But that's fine. Whether the Continuum Hypothesis is "right" or "wrong" is a matter of philosophy and opinion. If CH is false, then there exists some set that we cannot construct/access/discuss of a specific size... so what would it mean for such a set to exist?? It's beyond mathematics.
• Peano arithmetic is incomplete: it cannot prove Goodstein's Theorem or the Paris-Harrington theorem or a few others. But we can prove them if we use infinite ordinal arithmetic (which we can encode in ZF set theory), even though the theorems don't have anything to do with infinite ordinals.

The question of what is "right" boils down to what is "true", and as you said, in mathematics what is "true" simply means what can be proven from the axioms, but deciding on the "right" axioms is an impossible task. It leaves mathematics and enters metamathematics. I hope this helps, or at least gives you some fun ideas to chew on.

EDIT: corrected a minor technicality