In co-op, does enemy item/loot drop depend on who gets the kill?

Solution 1:

No, the player who kills it is unimportant. The relic affects global drop values so in theory even enemies not touched by the player (E.g. thresher's killing varkids in the caustic caverns) should have a slightly better drop chances. It is also worth noting that the Vault Hunter's relic only increases the spawn rate of greens and blues while decreasing the spawn rates of white items and even then, the change is miniscule. The relic is also unfortunately the only thing in game that can change these values (at the moment, anyway. Future DLC may change that). You really are better off equipping something else.