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The lorebook is where it is mentioned that their poms "change color depending on the flighty creature's current moods." I don't think there's any evidence of that in-game, so it doesn't explain the reason for their poms. I agree with the other poster about how the poms are possibly for attracting mates. When you first interact with a delivery moogle as a new player, they mention "dreaming about some beautiful mooglette with a rainbow colored pom-pom."
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Sorry for the necro and I know this isn't about their poms per say, but since i'm doing the moogle dailies now it reminded me about the one quest where you have to make some aether shielding for some moogles because they got overwhelmed by the aether coming from the crystals they're farming.