For those who are unaware, the cards in Yu-Gi-Oh! are all inspired by in-universe monsters from distant dimensions and ancient history. In the modern-day, duelists from Yu-Gi-Oh! compete with these cards in a game they call Duel Monsters.
Monsters can usually stand on their own, but duelists often fuse their monsters to overpower their opponents with strength or outsmart opponents with special card effects. The fusion monsters might end up being strong, weird, or even completely useless compared to their original forms but no matter what, monster fusion has left many Yu-Gi-Oh! fans with questions.
A fused monster card usually portrays two specific monsters who have fused into a stronger form, like Red-Eyes Dark Dragoon being the fusion between Red-Eyes Black Dragon and Dark Magician. Another card might show one of the same monsters fused to a different monster altogether, like Red-Eyes Slash Dragon not involving Dark Magician at all.
Yu-Gi-Oh! as a series has never stated whether fusions were permanent or temporary in the monsters’ stories' canon. It might be safe to assume that the contradictory fusions represent alternate futures that a monster might take. Alternate dimensions do exist in the Duel Monsters' canon, as shown in the art for Different Dimension Encounter where Warrior Dai Grepher’s two possible future selves meet in a strange void. Different Dimension Encounter might be the best answer to the question of contradictory fusions because it implies that many different timelines and different fusions occur concurrently.
Most fusions show two or more monsters fused into one, like Sangan and Witch of Black Forest becoming Sanwitch when fused. Fusion cards like Sanwitch are the literal fusion of two monsters, but the card The Dark Magicians displays Dark Magician and Dark Magician Girl merely teaming up for an attack.
Dark Magician The Dragon Knight is closer to a true fusion, with Dark Magician riding Timeaus, but he still hasn't really fused with the dragon. Team-up cards like these would obviously be stronger than any one of the monsters alone, but wouldn't the monsters be even stronger if they truly fused into one?
There are fusion cards that require a certain amount of monsters of a specific type to be summoned, like the Five-Headed Dragon that requires the fusion of any five Dragon-type monsters. The illustration for this card shows a dragon with five heads where one of them is clearly from Darkfire Dragon, but the duelist may use any five dragons to summon Five-Headed Dragon.
Since any dragons can be used to summon Five-Headed Dragon, it's entirely possible that the heads would take the appearance of whichever dragons were used to summon the monster even if it were something like the humanoid Dragonmaid series of cards. The art of Five-Headed Dragon might just be a possible example of what this monster would look like under the right circumstances, but isn't necessarily what Five-Headed Dragon would always end up looking like.
Dark Magician has 2500 attack points, and Flame Swordsman has 1800 attack points. A duelist would probably expect that their fusion would be a stronger monster than either of them alone. Not only does their fusion card, Dark Flare Knight, not have their combined 4300 attack points, but Dark Flare Knight has 300 fewer attack points than Dark Magician by itself.
Even Dark Flare Knight’s defense points are somehow lower than either Dark Magician or Flame Swordsman's defense points, and Dark Flare Knight's special effect isn't particularly worth summoning it for. At least Dark Flare Knight looks cool.
This list already covered fusions where the materials couldn’t possibly make the monster in the art, but there are plenty of fusion monster cards where the art clearly depicts fusions between specific monsters that the card's fusion materials don't list.
This happens across many card archetypes, but the problem is especially noticeable in the Frightfur line of cards. Frightfur Leo is obviously depicting Fluffal Leo fused with Edge Imp Saw, but Frightfur Leo can be summoned using any Fluffal monster even if it isn't Fluffal Leo. Duelists can even summon Frightfur Leo by using Fluffal Sheep instead of Fluffal Leo, despite Fluffal Sheep being the mutilated Fluffal monster in the art for Frightfur Sheep.