Yes, that does mean Azumarill can't double the power of its water and ice attacks. Its water attacks will be kind of weak but still get STAB, and its ice attacks will be too weak to be worth learning. (its super effective ice beam is weaker than its neutrally effective return) Even without ice attacks and with relatively weak water attacks, I think Azumarill can still pull its own weight in Emerald (especially when it's own weight is less than 300 N). You can catch Marill before the first gym, its fast growth rate makes it very easy to overlevel, and it can learn plenty of strong normal attacks (and water attacks) throughout the game. Also it's fortunately not fairy type. This is the moveset I would recommend.
Marill -> Azumarill @ silk scarf
Ability: huge power
- water gun -> bubblebeam -> surf
- tackle -> strength -> brick break (if you're okay with using TMs)
- whatever useless HM move no other teammates can learn