How can I assign more than 2 phone numbers to a contact on my iPhone?
I can add as many phone numbers as I want on the iPhone. When you're in edit mode in the contact, there's always a blank line under current phone numbers where you can start typing a new one.

If you click on the label of the number, you can select a label (work, home) and even a custom label if you scroll to the bottom of the label selection list.