The Azurill Hypothesis.
Myself and another poster on GameFAQs by the name of jayman7 discovered the most likely reason behind Azurill's many strange characteristics that aren't shared by its evolutions: it was programmed into Ruby/Sapphire using Cleffa as a template.
Cleffa and Azurill share evolution method, type, catch rate, gender ratio, and experience curve. GameFreak must have copied and pasted Cleffa's data when programming Azurill since they're both Baby Pokemon that evolve into members of the Fairy Egg Group. The evolution method (Happiness) wouldn't need to be changed, and the shared experience curve (800,000 Exp.) also facilitated the copy-paste job.
However, the other three characteristics they share are the ones that really stand out. The type (Normal), because it obviously doesn't match Marill and Azumarill's Water type. The catch rate (150), because Marill's catch rate is 190 which means Azurill is actually more difficult to catch than its evolution and that makes little sense. The gender ratio (25/75), because it doesn't match Marill and Azumarill's 50/50 and results in one-in-three female Azurill to evolve into male Marill.
As you can see, it's very likely that GameFreak took Cleffa's data and used it as a starting point for making Azurill. However, for some reason they didn't change enough. GameFreak didn't even adjust its type or gender ratio to match those of its evolutions, resulting in the anomaly that is Azurill.