According to [1], $\mathbb{R}$ is unique (up to isomorphism) "due" to its archimedian property. According to [2], said property is not firstorderizable.
This second statement makes sense on its own, when elements of the universe of discourse are real numbers, not sets of them. But to be able to speak about isomorphisms between different Dedekind-complete ordered fields one has to embed them into a larger theory. Why cannot this theory be a first order theory? ZFC can express the subset property, so where is the problem with [2]'s axiom 4?
The language used for the axiomatization uses the (non-logical) symbols $\langle +,\cdot,0,1,\le\rangle$ where $+,\cdot$ are binary functions, $0,1$ are $0$-ary functions and $\le$ is a binary relation.
The Dedekind completeness axiom states $$\forall U(((\exists x Ux)\land(\exists x\forall y(Uy\rightarrow y\le x)))\rightarrow(\exists x((\forall y Uy\rightarrow y\le x)\land(\forall z((\forall y Uy\rightarrow y\le z)\rightarrow x\le z)))))\quad (1)$$
Which is a second-order statement with $U$ quantifying over unary relations.
Call the other axioms $\Gamma$. The statements in $\Gamma$ are all first-order statements in the language $\langle +,\cdot,0,1,\le\rangle$.
What they're saying is that there is no first-order statement $(1')$ in the language $\langle +,\cdot,0,1,\le\rangle$ such that $\Gamma\cup\{(1)\}$ has the same models as $\Gamma\cup\{(1')\}$.
This follows from the Lowenheim-Skolem theorems, because a first-order theory cannot have a single infinite model unique up to isomorphism. On the other hand, the second-order theory $\Gamma\cup\{(1)\}$ (where $(1)$ is the only second-order statement and it merely quantifies over a single unary relation) has a unique model up to isomorphism.