The coffee you can get at no extra charge is at the drink station on the pool deck, and in the dining rooms. It is the regular Disney/Nescafe type coffee. There is never any extra charge for refills, so you don't need a mug program for that.
There is, as far as I recall, no free or reduced price refill program for specialty coffee drinks from the Cove Cafe, although they have had those cards where you bring it back and once you have purchased a certain number of beverages you get one free. My friend has used those. I don't drink coffee and they don't include booze so I've never had one of the cards.
They also don't sell coffee mugs for the station, except for some of the specialty cruises (Alaska had them), so bring your own if you want an insulated mug.
I bought the Alaska mugs, and they were not good for free refills from the Cove.