Math has it right in this case. There are no true definitions of these terms, only conventions. The meanings of these things differ by context. Things are the same, identical, or equal according to the 'equivalence relation' appropriate to the context in which one is speaking.
Famously Leibniz defines identity as having exactly the same values for all the same properties, and having no other properties. But that is in the context of propositional logic, where things have properties. It is the equivalence relation for that domain.
It does not match Kant's definition for identity, which allows for continuity across time, because age is a property, and two instances of you from different times have different ages. But the context is different. Kant is talking about beings, and so there is a different equivalence relation.
Your sets of parts are no different. How you declare their identities depends on the intended use. If an older part has more oxidation, and is of lower quality in some important way, the two are not identical. But if the point is just interchangeability in normal situations, they probably are identical in the relevant sense.
The infamous grade-school question is whether 1 = 0.999999... This is very important once you realize that 1/3 is 0.333.... and three times that is 0.9999... Clearly, formally, as strings of digits, these are not equal, but the equivalence relation between real numbers saves you. Two real numbers are equal if they differ by zero. The difference between these is less than 1/10^n for any n, so it can only be zero. So our two favorite definitions of the reals, as infinite sequences of digits, and as the points on the number line are not the same unless you create an explicit equivalence relation.
This seems like a special case, like the difference between integers, for instance does not require such a hack. But we forget that there are no integers in our universe. There are numbers of cars, and of fingers, and of... And we make an equivalence relation by finding pairings between the set we want to count, and some other set we have already counted. 'Bijection' is the equivalence relation between discrete numbers (of all sizes, even infinities.)
There is no proper, best, or only definition of any of these terms. They are all different words for being alike in specific ways, and those ways are relative to the intended purpose. So trying to discern them in any abstract sense is a misunderstanding.