RAW? I can't seem to find anything definitive. Looking at the spell, and the nature of it, I assume you take the same type of damage. Again, I don't see anything in the rules.
This, like every ambiguity, comes down to the DM.
Thematically, you have this magical bond and as your target/ally takes a sword across the face, you feel the slash across your face as well.
The only case I could make for it being just straight damage (other than the DM just wants that) is that in the spell, it says:
While the target is within 60 feet of you, it gains a +1 bonus to AC and saving throws, and it has resistance to all damage.
As for resistance, The target resists the damage type because of the spell. You are hit with half of that damage. Resistances don't stack (see other answers and PHB 197), but since this is brand new damage to you, you have not resisted it yet. If you are resistant to that type, you take 1/4 of the original total. If you are immune to it, you take zero.
- Ogre hits your ally with Warding Bond for 32 bludgeoning damage.
- Your ally (due to the spell) is resistant to that damage; he takes 16 bludgeoning damage.
- You take the same amount of damage, 16 bludgeoning damage.
Since you are a barbarian cleric, you are resistant to that damage while raging. You cast the spell then raged, thus are resistant. You take 8 bludgeoning damage.