The Sprens being small parts of the Honor-shard makes allot of sense. When Odium defeated Honor he probably had the choice to take the honor-power for himself, but that would have changed him to no longer being Odium.
So he instead decided to splinter the honor-shard instead, creating the Spren. That way he thought to make sure that honor could never again rise to challenge his power.
Another thing I came to think of: Perhaps the Radiants betrayal (them being honors champions in a way) weakened Honor in some way and that made it possible for Odium to defeat it?