Equality notions in logics other than first order logic

Question

Here https://terrytao.wordpress.com/books/analysis-i/ Terence Tao says:

However, the axioms provided are the standard axioms for equality in first-order logic, which already suffices for most mathematical applications (though one occasionally does have the need or desire to work in logics beyond first-order logic, in which case one may need a different set of axioms for equality).

What other notions of equality could he have in mind? Is the notion of equality in logics other than first order logic different from the notion of equality in first order logic? What are some axioms systems for equality of other logics?

Answer 1

He is alluding to the fact that in High-Order Logic, the equality axioms can be proved from the definition:

$∀F(Fx ↔ Fy) → x=y$

usually called The Principle of Identity of Indiscernibles.