Pink somehow broke/cracked her pearl, so White used it as an example for her not to fuck up anymore and White'd it. She gave her her own pearl since why would she even need one?
White stayed obsessive and focused on everything she did, just like White, instead of being bouncy like Pink. Pearls reflect their Diamonds, so White wanted her diamond to have Pink straighten up. Instead, her Pearl was so dedicated to her mission of "Protect Pink", she fell super in love and tight-lipped for her to the point where she betrayed her original diamond.
Pearl really is one of, if not the, most multi-faceted char in the show, next to Steven. Everything she did worked in like, 3 or 4 thematic ways.