"Subject to the fare rules" just means that the Companion discount code doesn't change the flexibility or refundability of your ticket purchase.

If you buy a cheap discount fare and a $99 companion fare, both tickets are nonflexible and non-refundable. Changes will incur a $75/$100 change fee per person. You can choose to pay the change fee to refund your money, but no replacement companion code will be issued, meaning you effectively forfeit your companion fare. You can avoid this forfeiture by only changing your ticket, never cancelling it outright.

If you buy a Full Flex Y fare or full price paid F fare, and apply a $99 companion fare, both tickets are flexible and refundable. Changes are free (just pay any applicable fare difference for the first person; the second person's fare remains $99). You can cancel the tickets at any time and get a refund, but again no replacement companion code will be issued, so once again this is an effective forfeiture of the companion fare. To avoid losing your companion fare, always change your ticket, never cancel it.