What does identity mean in first-order logic?

In first-order logic, identity is a fundamental logical constant expressed by the symbol "=". It represents numerical identity, a relation that a thing bears only to itself. Unlike descriptive identity, which refers to exact duplicates or copies, numerical identity signifies that if $x$ and $y$ are identical, there is only one entity that is both $x$ and $y$.123

The logical properties of identity are codified by two primary axioms:2

• Reflexivity: $x=x$, meaning every object is identical to itself.2
• Leibniz’s Law (Indiscernibility of Identicals): $(x=y)\supset (Fx\equiv Fy)$. This states that if $x$ is identical to $y$, then any property $F$ true of $x$ must also be true of $y$.2

These axioms imply that identity is symmetrical and transitive. While identity is often taken as a primitive concept in first-order logic, in higher-order systems it can be defined using the principle of the identity of indiscernibles, where $x=y$ is defined as the condition that every property possessed by $x$ is also possessed by $y$ and vice versa.24