It is often implied that the ordered pair property (OPP) $$(x, y)=(z, w)\iff x=z\textrm{ and }y=w,$$ is sufficient to define an object with the intuitive properties of an ordered pair. One can argue however that the OPP only establishes that $(x, y)$ and $(y, x)$ are different objects (when $x\ne y$), not that $x$ and $y$ can be placed in some order. In ZF theory it seems that Kuratowski pairs $$(x, y):=\{\{x\}, \{x, y\}\}$$ succeed only because of the special property $$\bigcap (x, y)=\{x\},$$ which allows $x$ and $y$ to be distinguished and one arbitrarily declared first. For Hausdorff pairs $$(x, y)=\{(1, x), (2, y)\},$$ the order is imposed by the Von Neumann integers.
Question: Is the OPP by itself enough to define ordered pairs generically or is an additional property needed? If so what is it?
Yes. Even with only the OPP, we can define a projection to one of the components (declared the first component) and break the symmetry - we do not necessarily know if the projection picks the "originally" first.
Assume we have any way to determine a pair object from given inputs, i.e., a class function $\Phi$ that defines ordered pairs. This means that $\Phi$ has the properties $$\tag1\forall a,b\exists!p\colon \Phi(a,b,p) $$ and $$\tag2\forall a,b,c,d,p\colon \Phi(a,b,p)\land \Phi(c,d,p)\to a=c\land b=d $$ Then we automatically can define $$\tag3 \Psi(p,x)\equiv \exists y\colon \Phi(x,y,p)$$ This from $(2)$ and $(3)$ one immediately finds that $\Psi$ is a partial class function, i.e., $$\tag4 \Psi(p,x)\land\Psi(p,u)\to x=u$$ and that it is defined whenever $p$ is an ordered pair. Apparently, $\Psi$ maps each ordered pair to one specific (call it "first") of its components.