There are nice examples in geometry. For example, the Euclidean plane can be characterized on the one hand by Hilbert's axioms, on the other hand by axioms for complete ordered field $+$ vector space $+$ inner product.

Another example is that of a Pappian plane, characterized on the one hand by the three projective plane axioms $+$ Pappus theorem, on the other hand by the field axioms (from which the plane may be constructed via homogeneous coordinates).