I was referring to drinks like those served at Starbucks, open containers, lidded drinks like fountain beverages etc.
As another pointed out upthread, if the coffee was transferred into a sealable container (like, say, an empty water bottle) you could conceivably tuck it into your bag and bring it on board.
As for carrying things in ones hand on a coaster...not a good idea. Be it a cell phone, camera or a bottle of water.