To intercept a missile you need to intercept it - that's it. You don't need to be traveling the same speed, or have the same mass, you just need to be physically where it is at the time it's there.

ICBMs aren't armored - and they can't be. They're spacecraft of a sort, and doing almost any amount of intentional damage will prevent them from functioning. Nuclear warheads are hard to detonate - the behavior if disrupted is the weapon likely goes inert because you've ruined the implosion mirror or timing circuitry.

The problem is that interception is hard because your engagement times are very short, and you're off-trajectory. Initial deviations in your aiming create larger and larger errors at the target, and any sort of maneuverability on the warhead's part means you're unlikely to be able to correct in time - hence the focus on trying to hit missiles in the boost phase, when they're by necessity more predictable.