Doubling Rings came out in the CRB. They make dual-wielding cheaper, the fundamental runes of one of your weapons are duplicated to the weapon held in the other hand.
Blazons of Shared Power seem to do the same thing, in a slightly different way.
The differences seem to be in fluff. You do not have a limit on how many rings you can wear, so replacing the two rings with brass emblems is meaningless mechanically.
Both work with one-handed weapons wielded simultaneously, neither works with thrown weapons.
So why would I pick one over the other?