A dating simulation game
I say "targets" because all the men you end up in relationships with are billed as "straight" and Haru is an overly-energetic, horny puppy who admits he partakes in "sexual harassment" as a work pastime.
These games are often designed to be played multiple times.
Tools like multiple save slots, quick-save features, and the ability to skip text and rewind help players efficiently pursue every potential partner character and reveal every story path.
It’s like backwards-engineering a narrative database -- and you may be surprised to find how much the story changes when you focus on a different partner each time. Hakuoki is a period piece, taking place in the Bakumatsu period of Japan at the time of civil unrest between the Emperor and the Shogun.
These dating sim elements are often referred to as Romance Sidequests in the RPG genre.
Life Simulation Games also tend to feature dating sim elements.
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This creates a very different relationship dynamic, since you are inside Haru's head and are the active pursuer, rather than the pursued.