In Shopify "Customers" section there is no concept of "Billing" or "Shipping" address. There is just the list of Addresses attached to each customer.
The Billing/Shipping address is distinguished only when you use the address in the Order - so that the Order level this address needs to be put either in the Billing or Shipping address fields.
While with Shopify CSV import of Customers you can import just one address, you can import several addresses using the Excelify.io app (https://apps.shopify.com/excel-export-import).
If you need help with that, here you can contact us to assist with that import: https://excelify.io/contact-us/
Nick here from Shopify.
Just to confirm what Maris is saying. I double checked with our higher level technical support team for you on this, and it is only possible to import one address via a CSV. Looking into different apps would definitely be your best bet moving forward. Another app to look into for this other than the one Maris mentioned is Cart2Cart Store migration which you can see here.
All the best, Nick
I know this thread is a couple of years old, but I figure someone should update it since I ended up here.
I'm just attempting to import customers with multiple addresses and found this link that describes how it can be done via the customer CSV file: