I think they have tens of thousands of names, addresses, phone numbers, email addresses, configuration details and more in a data base.
I think when they look things up they can do it either by name, phone number or ZIP code along with any other field.
Since ZIP code has the least number of digits you have to verbalize to the call center operator, they asked for that one. Other call centers for different companies ask for phone numbers, some account numbers, some by name. Depends on how the call center is set up.
ZIP codes matter for delivery (dealer assignment), how they ask you at the call center to find your record, not related to delivery time at all.