Long ago, pawns were only allowed to move 1 square at a time (side note: fianchettoing was therefore very common then, as playing e3 or d3 opens the diagonal for one bishop, only to block the other!) ; the rules were modified to allow pawns to move two squares ahead on its first move in order to speed up the game.
Now, this had some major impact strategically because then players could move their pawns 'past' their opponents' and avoid capture, which would have resulted in the need for a major re-think of chess strategy, therefore the en passant rule was implemented to maintain a good level of similarity to the 'old' game.
One possible reason why this rule wasn't expanded and implemented for pieces is due to the static nature of pawns (which can't move backwards), which is not the case for pieces; having a pawn move past a square defended by an enemy pawn to evade capture was deemed unacceptable, whereas it is alright to evade capture by a piece. The strategic implications of the former is extremely severe and would lead to a completely different game, but in the case of the latter it's lighter and the nature of the game remains somewhat unchanged (perhaps because the piece can still relocate to 'chase' the pawn?).
But in any case, "rules are rules", however arbitrary they may seem (don't hope to change them, especially on the basis of "it doesn't make sense!"); to digress: why is the queen the most powerful piece? (Further digression: https://www.youtube.com/watch?v=N_AhNHrfu4Q). Having said so, questioning the motivation of these strange rules is certainly not unreasonable, and hopefully the above gives an idea over the rationale of not allowing pieces to capture en passant.
On a trivial note: it was actually suggested at a FIDE conference relatively recently (clearly by someone who didn't know chess very well!) that pieces be allowed the capture en passant. Needless to say, the suggestion was not taken seriously!