So... is Penguin Musume♥Heart finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: November 14, 2008
Manga finished: January 19, 2010
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Penguin Musume♥Heart follows Sakura Nankyoku, a dedicated otaku who unexpectedly becomes the Student Council President of her school.
The anime chronicles Sakura's chaotic life and the various humorous situations she finds herself in after assuming this new role.
Expect a series of offbeat and lighthearted adventures involving Sakura, her eccentric family, and the diverse personalities within her class, as they all navigate the ups and downs of school life.