User becomes stronger, faster, more durable, etc. when they come into contact with a moon, moonlight, or lunar energy/substance, possibly unlocking abilities related to the affinity and enhancing their existing powers. Some users may be able to draw sustenance from the moon or even slow or stop aging.
Tsukiyo (Boruto: Naruto Next Generation); the strength of his Moon Shadow Technique is determined by the amount of moonlight in the area with his skill having its maximum potential under the light of a full moon
Enshin (Akame Ga Kill!)
Saiyans (Dragon Ball series); via their tails
Engi Threepiece (Dream Eater Merry)
Werewolves (Rosario + Vampire)
Sailor Moon (Sailor Moon)
Naruto Uzumaki (Naruto the Movie: Guardians of the Crescent Moon Kingdom) via Crescent Moon Rasengan
Werewolves (Shadow Falls); the strength of their powers vary with the moonphases
Chameleons (Shadow Falls)
Moonstone Bracelets (Ah! My Goddess)
Moon Rings (Mako: Island of Secrets)
Trident (Mako: Island of Secrets)
Moon Pool (H2O: Just Add Water/Mako: Island of Secrets)
Saïx's Claymores (Kingdom Hearts)
Silver Crystal (Sailor Moon)
Moon Tiara (Sailor Moon)
Arethusa (Valkyrie Crusade)
Waterbenders (Avatar: The Last Airbender) are at their peak during a full moon.
Through the power of Khonshu, Moon Knight (Marvel Comics) received increased strength at night and under the light of the moon.
Saïx: (Kingdom Hearts II) "Do you feel it? The moon's power."
Umbra Witches (Bayonetta) are able to walk on walls and ceilings under the moonlight.
Enshin's (Akame Ga Kill!) Teigu's, Moonlight Sword Dance: Shamshir, aerial blades power is dependent on the lunar phase, with full moon making them strongest.
Arethusa (Valkyrie Crusade) gets power from the moon.
Known for their speed, Werewolves (Rosario+Vampire) become faster and generally more formidable the brighter the moon shines.
Naruto Uzumaki (Naruto the Movie: Guardians of the Crescent Moon Kingdom) infusing his Rasengan with moonlight to create the more powerful Crescent Moon Rasengan.
Under the full moon, Tsukiyo (Boruto: Naruto Next Generations) using his Moon Shadow Technique which manifests the target's shadows as tangible clones that are able to use all the same raw might, techniques and abilities as the original.
Carrot (One Piece) transforming into her Sulong form under the full moon, giving into their feral instincts and gaining enhanced physical abilities.
Pekoms (One Piece) transforming into his Sulong form by seeing the full moon becoming a lion beast and feral instincts and enhancing his physical strength.