Moe: Ami and Yumi are this. Atchan and Julie can also count.
Paranoia Fuel: Imagine a fan with hypnotic eyes stalking you, just like Harmony did to Ami, Yumi and Kaz.
Retroactive Recognition: Downplayed exampled, but even before this series and their singing the theme to Teen Titans, some viewers recognized their works from, of all thing, Guitar Freaks imports where some of their songs are prominent in that game. A few if them even show up in a few episodes.
So Okay, It's Average: The show is mostly infamous because of its marketing campaign and coming out of the blue without much warning. On its own, it's a fairly average cartoon show.
Superlative Dubbing: The excellent Spanish dub was the reason of its popularity in Latin America, with Karla Falcon as a snarky Yumi and Leyla Rangel as a overly cute Ami and the good chemistry and comedic timing they had.
Tastes Like Diabetes: The color scheme and overall "cuteness" of the series is likely to turn away a lot of viewers.
Unexpected Character: The Talent Suckers' cameo appearance in "Puffy B.C.". That episode took place entirely in the pre-history and they only appear because of a never seen before (or since) Time Machine.
We're Still Relevant, Dammit!: The show hit a twofer with this. Puffy AmiYumi were way out of their peak popularity when the show landed, and it also came in the dying wave of anime-parody/anime-referencing western toons.