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I set up the billing information in the checkout settings :
Home » Administration » Store » Configuration » Checkout settings
Enable profile copying on this checkout pane
and copy from Shipping information.
There are a couple of strange behaviours:
-First If option "Hide the country when only one is available" is selected, and the shipping information fields are empty and the option to copy the fields on the billing information is selected, if the user submits the empty form an error message is produced saying that the shipping fields are required but an entry in the billing information address book is created with just the default country field.
-Second, every time the user "ticks" and "unticks" the "My billing information is the same as my shipping information." and a default address is copied over from the shipping information pane, a new entry in the billing address book is created. I end up with many entries in the billing information address book, all with the same repeated address.