On Remote, the Billing contact is the company user assigned to receive email notifications about invoices and invoice payments. You can have more than one Billing contact. A Billing contact is an assignment and as such, a company user with any role can also be assigned to be the Billing contact. To make a company user a Billing contact, all you need to do is to edit their role.
See also: Can I change a Company user's role?
On Remote currently, every company admin receives all the notifications, but if you want the Billing contact to only receive emails related to billing, then they will need to be made a Company viewer. This will also limit the number of actions they can carry out on Remote.