I believe that what you are describing is a shortening of the possessive, such as Sammy's baseball. In this case, they are indicating a place of business, such as Sammy's Pizza Parlor". They are further shortening it by neglecting to say the kind of business, so Sammy's Pizza Parlor simply becomes Sammy's.
It's not as crazy as it might sound. If there were two pizza parlors in the neighborhood, Billy's Pizza Parlor and Sammy's Pizza Parlor, when asked where you were going for lunch, you might just respond "I'm going to Billy's." Everyone would know what you meant, and shortening the name of the restaurant that way would demonstrate familiarity with the neighborhood.
Business names like the ones you describe are trying to advertise to potential customers that they are familiar parts of the local area.