I know gingerbread for a long time was made any time of year as kind of a show that you could afford the spices, but somewhere it morphed into a Christmas thing. I'm sure the same thing happened to peppermint/candy canes and plenty of other foods and flavors. I'd love to hear about anything you know about I haven't mentioned, too.
I asked some friends and they said other things like mince meat pie and goose or duck, for more ideas.