Not only people possess. Things possess: the land's resources, the moon's gravity. In this case, today "possesses" many things, including a lesson that didn't occur yesterday and won't occur tomorrow. Ditto tomorrow. Even abstract nouns can possess: e.g., "Freedom's ring."
In a logical arrangement, you can think of the possessed object as a subset of the possessor. John might be just John, or he could be the set of John and all his things, and John's hat is an element of that set, the hat of John, the hat sub-object that belongs to the set of John and his things. Hence, the lesson of today, today's lesson, to world of tomorrow, tomorrow's world.
if by it's very nature possessive only applies to those who possess
If that were the case, we'd have his and hers but not its.