While operators are 'overloaded' all the time, equality is a bit of a special case as it is part of logic rather than some algebraic operation.
In model theory you don't require your models to have an equality operator, they have one simply by being logical constructs.
Then again mathematicians use quotient spaces so transparently that you might as well consider:
5 = 1 (mod 4)
as 'overloading the equality operator' even though the technical definition implies that those 5 and 1 are different from the 5 and 1 in the set of natural numbers, and in Z/4Z the symbols 1 and 5 refer to the same object.
In model theory you don't require your models to have an equality operator, they have one simply by being logical constructs.
Then again mathematicians use quotient spaces so transparently that you might as well consider:
5 = 1 (mod 4)
as 'overloading the equality operator' even though the technical definition implies that those 5 and 1 are different from the 5 and 1 in the set of natural numbers, and in Z/4Z the symbols 1 and 5 refer to the same object.