Observe that when you identify x with y, you are already drawing an equivalence between their formal representations (i.e. x, y), and the rule holds only for f

Observe that when you identify x with y, you are already drawing an equivalence between their formal representations (i.e. x, y), and the rule holds only for f that is blind to the formal representations, and distinguishes only between these equivalence classes. But within math topics there are notions of equivalence that are stronger than just that that the topic is uninterested in "piercing through the equivalence veil". For example, computer scientists often talk about Turing machines being equal even when their alphabets are disjoint, as long as their structure is the same, reserving equivalence typically to mean accepting the same language.