Warnings: Contain spoilers of the game and possible future spoilers of the manga.
Game basic storyline (that appears to be followed by the manga): The story of Heart no Kuni no Alice is in fact an adaptation to the novel "Alice in Wonderland", with some adaptations to fit in an Otome game (in this case, she is surrounded by good looking guys that slowly become in love with her).
Truth about wonderland: That world is actually, the world of clock and time, the reason why there is 12 wonderland role-plays is because each of them represent an hour of the watch. That is the reason why each of them say that their life is meaningless and can be replaced, and thus why each one of them have a clock that brings them to existence (they have them in places of a heart, and even if they die, as long as the clock is not destroyed, they can be reborn (or as they refer, replaced)), without the watch, they don’t exist.
This is also the reason why they all love Alice, as an outsider/human, she is the one that gives them meaning by living those hours.
Relationship between Alice and the White rabbit: Thought Alice doesn’t understand, she was the one that constantly brought the white rabbit into her mind, causing him to fall in love with her before she came to wonderland. Alice remembers dearly her Sunday afternoons with her sister, which is why she decides to spend each afternoon on the garden remembering those days. Well, as explained before, Peter White, like the other 12 characters, represents a clock hour, and he is in fact, the hour that Alice used to spend with her sister on the Sunday.
This makes Peter be remembered repeatedly, and each time she remembered that hour he was brought in the real world and visited with Alice over and over again. Making Peter create an obsession towards Alice, although it is true that he is loved by Alice, he is loved in his meaning (the time spend with her older sister) and not in his form (the white rabbit).
Alice in the manga however still doesn’t know why Peter loves her
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In the manga, he is also uncaring towards other, but does not show the complete repulse that he shows on the game, so I shall assume that the others existence to him are meaningless to him.