Probably, Johannes Ebert is right: (almost) all natural mathematical objects may be characterized by a universal property. The question is now what we understand exactly by the the fact that universal property is delving in the concrete habitual definition.
More concrete, let consider the usual definition of a factor structure, let say a factor group (of $G$ modulo a normal subgroup $H$). There is also a universal one: A factor group is (up to an isomorphism) an epimorphism (i.e. a surjective group homomorphism) $G\to G'$. Does the second definition delve the first? I really don't know!
Another example: Having two $R$-modules, $M$ of the right and $N$ of the left, one may define the tensor product as a factor of the free abelian group with the basis the cartezian product $M\times N$ modulo the relations which emphasize the bilinearity. Secondly, we may define the tensor $M\otimes_R-$ as the right adjoint of the functor $Hom_R(M,-)$, definition which may be extended for $M$ in a cocomplete abelian category. This time the possibility to change the settings leading to a more general definition stands as an argument that the universal definition is not delving in the usual one.