I'm wondering how this would differ if one were trying to say "My son is Omar and my husband is Mahmoud."?
Technically correct, but most people would assume the duolingo response (i.e introduction: here is my son omar and my husb... etc) I would say (for your sentence): اِبْني هُوَ عُمَر وَزَوْجي هُوَ مَحْمود. By adding the ' هُوَ ', which means he is, you basically say My son, he is Omar and my husband, he is Mahmoud. This is used when like, maybe, pointing out from a list of names, etc.