If seasonal were used as an adjective it would mean: of, relating to, or characteristic of a particular season of the year. Example: We have a selection of seasonal fresh fruit on sale today. It could also mean: fluctuating or restricted according to the season or time of year. Example: There are companies whose markets are seasonal. Seasonal Rainfall.
But in Season's Greetings, season's isn't an adjective. It's a noun. Greetings of the season.
Yesterday, the 26th, I heard a singing Christmas display in a store.