While it certainly could be an enforced cap by design (dynamic shadows are "expensive"), this sounds like it could be vsync as well.
Basically (and inaccurately), there is a frame which you see, and a frame which is drawn on. When the drawn one is complete, the contents are passed to the screen for you to see while a new frame is drawn.
In the "olden" days, video was drawn line by line at the upper corner and moving to the lower corner and then there was a delay when the "cursor" reset to the upper corner. The reset happens during a "wait state" (aka swap interval, vertical sync command etc) and the frame would be swapped during this very brief window.
vsync ON means wait for this to swap frames. You get FPS capped at the refresh rate of the monitor, a very stable picture, but fast movement feels choppy.
vsync OFF means don't wait. You get FPS capped at the computer's ability to render it, but you get frames being swapped out while the image is partially rendered (aka "tearing"), but fast movement feels smoother.
With vsync on, if you can't meet or exceed the refresh rate, then the hardware has to pause and wait for the next cycle. This means you wait 2 frames-worth of time to swap. In games, this tends to happen in extended bursts because large amounts of staffage, explosions, etc. taxes hardware. The result is the FPS appears to drop from locked 60 to locked 30.
Plenty of psuedo-technical info here, but that's the gist of it.
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.
Best Answer
Do you have PhysX set to higher than 'low' in your video settings? If yes, try setting it to low as you might have a large performance hit due to the fact that the PhysX would be CPU-bound if you have an AMD GPU. (Source: PC Gaming Wiki; Borderlands 2 - PhysX)
I'd also try setting the framerate option to 'Smoothed 22-62'. What it does is try to maintain your frame rate between 22-62, automatically adjusting the engine (graphics, effects, draw distance, etc) on-the-fly, to try to maintain this frame rate in parts where your PC starts to drop the frame rate below the minimum (22fps). Source: Steam forum post on 'Capped 60, smoothed 22-62, or unlimited+vsync on?' thread
If that still doesn't work for you, try changing the framerate setting to 'Unlimited' instead. Then, try changing MinDesiredFrameRate from 35 to 60 in WillowEngine.ini in the USERS directory:
C:\Users\%USERNAME%\Documents\My Games\Borderlands 2\WillowGame\Config\. It may result in the expected performance from high-powered hardware. (Source: PC Gaming Wiki; 'Borderlands 2 - Microstutter on Powerful Rigs')