Perhaps they mean to imply that they are wishing you only a happy New Year's Day , not a whole year?

Just a wild stab in the dark.
Some Americans (and I stress the 'some' advisedly) tend to make possessives out of a lot of things which are not so. I once worked in an office building which had, on the ground floor, a store called The Agora. Agora, of course, is the Greek word for marketplace. There was a co-worker (more than one, actually) who kept refering to having bougth somewthing "at Agora's." I suspect he thought Agora was the proprietor's name.