So... is Azumanga Daiou The Animation finished or not?
YES, it is finished.
Why?
The anime finished after the manga:
Anime finished: October 01, 2002
Manga finished: March 21, 2002
MangaUpdates contains information about the anime adaptation and indicates that it is complete.
Watching Order (by release date)
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Synopsis
Azumanga Daioh The Animation follows the daily lives of a group of high school girls, led by the precocious child genius Chiyo, as they navigate the challenges and joys of adolescence.
Each girl has a distinct and quirky personality, from the energetic Tomo to the gentle giant Sakaki, and they are all guided (or sometimes misguided) by their eccentric teacher, Yukari.
The anime focuses on their interactions with each other, the humorous situations they find themselves in, and the small, relatable moments that make up their high school experience. Expect slice-of-life adventures filled with plenty of comedy and heartwarming moments, showing a realistic, yet exaggerated, portrayal of their school days.