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To add the Customized Address Label Name in the Core, follow the steps below:
How To Add The Customized Address Label Name In the Core
Go to the Setup Module.

In Setup, navigate to Configurations and then click on the Payments Tab.

Whilst in Payments, scroll downwards to the Online Configurations Section.

Enter the display name for your Customized Address, such as Home, Billing, or Shipping.

The Name entered in the Address Name field will appear on the Widget and Customer App as the Label for the Customer Address.