Is this relational characterization of equality in Wikipedia accepted?
The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and transitive. The relation of equality is also antisymmetric. These four properties uniquely determine the equality relation on any set S and render equality the only relation on S that is both an equivalence relation and a partial order. [My italics]
If one follows wikipedia, antisymmetry is defined in terms of equality, hence the definition is circular (impredicative?):
In mathematics, a binary relation $R$ on a set $X$ is antisymmetric if there is no pair of distinct elements of $X$ each of which is related by $R$ to the other. More formally, $R$ is antisymmetric precisely if for all a and b in $X$
if $R(a,b)$ and $R(b,a)$, then $a = b$, or, equivalently,
if $R(a,b)$ with $a ≠ b$, then $R(b,a)$ must not hold.
In posets antisymmetry is typically defined in terms of isomorphism, not equality, particularly when categorified. In any poset, such as $\bf 3$, itself a category (as well as an object of $\bf Poset$) distinction between objects should only be resolved up to isomorphism, not equality - the latter is considered an evil property. Regarding this, the n-lab entry on partial-order includes the statement:
the antisymmetry law, saying that $x \leq y$ and $y \leq x$ imply $x=y$, is evil in a certain sense. (On the other hand, it is not evil if taken as a definition of equality.)
So I wrote "typically" above to consider the parenthetical. Of course the objects of $\bf 3$ are not equal, but they are also not isomorphic (would have to look in pre-order categories. Here I'm focusing attention on the relation between equality and isomorphism).
Note that the n-lab entry on equality considers "different kinds of equality" complicating the issue.
By substituting $\cong$ for $=$ in both Wikipedia entries, doesn't isomorphism also yield both an equivalence and an order relation?
NOTES: it's interesting for such a basic, fundamental relation to be so convoluted, see eg Barry Mazur's definition in "When is one thing equal to another thing" ("up to unique iso") and Andreas Blass et al's "When are two algorithms the same". Historically, the issue also intrigued Leibniz who proposed the principle of identity of indiscernibles. Physically, nature at low quantum numbers seems to obey this principle: we cannot tell apart any two protons say, or water molecules. But already there are combinatorially ~10^200 possible molecules of less than 50 Daltons composed of H, C, O, N, S and maybe a couple other elements (~10^120 times more than the estimated number of atoms in the universe).
SARCASM: perhaps equality is unique up to unique isomorphism?