Both approaches are linguistically sound, but only one is accepted as a standard; namely, "my wife's and my". Standard forms are chosen somewhat arbitrarily. This means that they don't have some sort of objective "correctness"; it also means that you can't argue for the correctness of a non-standard form based on logic. There are many logical ways to convey ideas, and one was chosen to be the standard. If you wish to communicate in a context where adherence to formal/standard rules is beneficial, then you should choose the standard form.