Looking at the etymology, impeach is from old-french "empecher", and "em" was an alternative form of the old-french prefix "en".
And "empecher" is itself is derived from the late latin "impedicare", which uses the Latin prefix "im" from which the French prefix comes. And is prefixed to "predica".
So it it is a prefix.
Of course, the latin (and French) prefixes aren't used to indicate opposite meaning, like "in" often is. But that's just yet more bullshit.
To be fair it was a prefix in the Latin word that it's ultimately derived from. We still treat it a little like a prefix when we use the im- part of the related "import" like a prefix, as we also have "export" and "transport"
Ehh, at least we don't randomly assign every noun one of three genders, and have to memorize them all in order to use the correct form of "the" when speaking about it.
As a slight tangent, a similar peculiarity in English (which I don't know of a name for) is where you can use the opposite words for similar actions, e.g. you can chop a tree down and then chop it up.
Reminds me of folding cardboard boxes. If you are taking a flat piece and make a box of it, are you folding a box or unfolding the cardboard. Or both. And when you do the reverse, you do the same, do you not?
Not exactly the same, but that also reminds me of autoantonyms or Janus words. The word dust can be used to describe adding dust or removing dust, for example.
I actually use "shevelled" alongside many other words which to my mind "should logically exist" - for example, at the weekend I dismantled and then remantled a wall in my garden.