With the old Authenticator there was a way to extract a serial number of the form US-NNNN-NNNN-NNNN and a 10 character restore code.
These were the standard mechanism I used when transferring the authenticator from one phone to a new one.
How is this done with the new mobile app? I cannot find any means to extract a serial/restore code from it to help with the transfer to a new phone.