Here's how I think Alchemist's Fire works "officially":
1) on attackers turn: take 1d6 fire damage
2) on target's turn: take full-round action to extinguish (Refl DC 15) or take additional 1d6 fire damage
3) target's 2nd turn: A.F. has burnt out. If not extinguished in last round and you are wearing flammable materials, make a Reflex saving throw DC 15 to avoid having those catch fire.
The slimy fish-man wouldn't need to worry about the last part, so only takes 1d6 or 2d6, depending on whether he wants to spend a full round to extinguish the A.F. However, someone wearing flammable clothing (or equipment!) would need to worry about those catching on fire.
Someone only *splashed* with A.F. takes a single point of fire damage on the attackers turn. Then he skips straight to point 3): Reflex save to avoid having your gear catch fire.