When I was in India, if an Indian classmate asked to have a sip from my water bottle, they would often place their thumb between the rim and their mouth and pour the water, rather than sipping directly from it. They said it was so I wouldn't have to "drink after" them. I don't know if that is what this sentence intends to say, but it makes sense in that context.
Or you might be about to drink with the Dread Pirate Roberts, in which case drinking second won't save you because he has an immunity to poison anyway (:
Yes, it seems to be the norm to pour the water without touching your lips to the rim of the container*... sometimes with considerable distance between mouth and bottle ...I tried to copy that (I think it's very sensible) ...not easy to master that technic though.
*On the subcontinent that is