It is mainly so special because it can arise from many different openings, both open and closed, e.g. the Queen's Gambit, Nimzo-Indian, Caro-Kann and the Alapin variation of the Sicilian. It is simply much more common than a mirrored situation with an isolated pawn on e4, which would require exchanging the white f-pawn, which often stays on f2 to protect the king. In general, an early isolated pawn on the n-file requires an early n4 move by White and n5 by Black, followed by undermining pawn moves in the adjacent files. Those are most likely to happen on c-, d- and e- files.
An IKP (Isolated King's Pawn – an isolani is often called an IQP) would give the same advantages, but in addition:
- it would be harder to defend than an IQP
- assuming kingside castling, the half-open f-line would give White even more attacking chances, at the cost of being more exposed as well
A isolated c-pawn is even rarer, and has less influence on the center than a d- or e-pawn. All other things being equal, this is definitely a disadvantage for the side having an isolated pawn.
Hanging pawns are another commonly occurring structure where (potentially) weak pawns have compensation in the form of center control and space advantage.