I'm a bit dubious about your sweeping generalisation that "most mathematicians haven't don't any logic".
I also agree with GP that logic also does not deal with absolute truth. Proofs in formal logic may be applied to models, but they only provide truths modulo the assumptions made in those models.
As for your remark on equivalence. Your original description is one of syntactic identity, a sort of free equivalence. But any relation that is reflexive, transitive and symmetric can be considered an equivalence (indeed, there are infinitely many equivalence relations over the integers). Also note that this is not the same as how "=" is used in computer programs where for the most part it is asymmetric (a=b is rarely semantically equivalent to b=a).
I also agree with GP that logic also does not deal with absolute truth. Proofs in formal logic may be applied to models, but they only provide truths modulo the assumptions made in those models.
As for your remark on equivalence. Your original description is one of syntactic identity, a sort of free equivalence. But any relation that is reflexive, transitive and symmetric can be considered an equivalence (indeed, there are infinitely many equivalence relations over the integers). Also note that this is not the same as how "=" is used in computer programs where for the most part it is asymmetric (a=b is rarely semantically equivalent to b=a).