I'll have to test what happens when an enemy does it... however, I found this to be interesting.
Remember how the Dark Elf is weak vs Holy physical attacks and absorbs Holy magic attacks?
Well, the same occurs with Ice Shield+Fire Armor or Fire Shield+Ice Armor combos.. although the only test was using the Icebrand as a weapon on Cecil himself.

There was a clear damage difference (2x increase) with that combo equipped... testing the Icebrand strictly on the Fire Shield+Armor resulted in a 2x reduction of damage.. and testing the Icebrand on the Silver Shield+Armor resulted in no extra base damage. So, it does affect your characters as well. I'll have to test it with enemies... and test it with Weak+ as well and see if there's a difference.
It doesn't occur with enemy type weaponry vs enemy type armor though (as the character is obviously not of that type after all).