So... is Princess Lover! finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: September 21, 2009
Manga finished: November 27, 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
Teppei Arima, a regular teenager, experiences a sudden and tragic loss, the death of his parents, leading him to be adopted by his wealthy grandfather.
He is thrust into a world of unimaginable luxury and privilege, a stark contrast to his previous life.
Teppei is expected to take over his grandfather's powerful financial empire, leading him to attend a prestigious school to prepare.
He must navigate the challenges of high society, including the prospect of an arranged marriage with a wealthy heiress, as he adapts to his new, extravagant lifestyle.