No, you can't rephrase it as "Bigamy is having one wife in addition to that", because there's nothing for "that" to refer to. An English speaker, hearing that sentence by itself, would say, "Having one wife in addition to what?" You could rephrase the sentence as "Bigamy is the condition of possessing one more wife than the proper number", if that helps you understand it better.
"Having one wife too many" is maybe a little bit of an unusual phrasing as a whole, but all of the parts are common enough.
Having is just a gerund, as in "Having a safe place to sleep is important" or "I like having a car."
I assume you know what "too many" means! "One X too many" is a way of saying, "one more X than the proper number". For example, if you asked someone to bring you five shirts, and they brought you six, you might say, "No, that's one shirt too many."