No one but the customer is allowed to change their name in our org so it's super redundant to have two name fields.

When you go to Customer record, there is a radio button to choose from Individual or Company. When you select Company, you get single Name field, whereas when you select Individual you get Name consisting of 3 fields. This would be why you see 2 Name fields which are both mandatory.

Sorry, I'm seeing 2 name fields because there is one for company and one for individual? Or, I'm seeing two because this is an individual and one is for the '3-field-name' and one is for the composite of the '3-field-name'? So is there no way to have one name for an individual?

One for Company and one for Individual,exactly.