The phrase "to designate a companion" means to select or choose someone to be your partner, friend, or someone you will spend time with and have a close relationship with. Full definition
The pass allows you to designate a companion who flies essentially for free (plus a nominal tax) on any flight on which you are flying. (rewardexpert.com)
There, you can designate a companion online, or you can call and do so over the phone. (thepointsguy.com)
Once you have earned the required points, you can designate a companion who can then fly with you for only the cost of the taxes and fees of their ticket. (straighttothepoints.co)