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JGWeissman comments on Exponentiation goes wrong first - Less Wrong
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"Uniqueness" of the natural numbers means that for any two models of the axioms, there is an isomorphism between them that preserves the successor function and identity of zero. "Uniqueness" for second order logic would be similar, though I am less familiar with the formalization, so I won't list all the things the isomorphism should preserve.
If you manage to find axioms that capture your intuitive notion. The idea is, even if induction fails, there is still a "unique" notion of natural numbers, it just isn't adequately described using induction. When you are presented with a convincing argument for a given axiomatic definition not capturing your concept, you just find what assumptions led to the disagreement and change them to obtain a better description.