I plan on making a druid character who later accepts the curse of lycanthropy into his body in order to control it, something like a shapeshifter from Baldur's Gate. Am I able to use the multi-attack action and the bonus action I have to make a third attack (claw attack) as defined under the two weapon fighting style rules? I have yet to find anything saying whether or not natural weapons are light, as it seems common sense that a hand (claw) is considered light, as it is your hand.
PC lycanthropes don't have multiattack
The rules on PC lycanthropes are on Monster Manual page 207, and make no mention of the character gaining the Multiattack action. The characters are said to be proficient with natural attacks such as bites and claws, so the character can use those. The Multiattack is conveniently replaced by the PC's own extra attack from their class, if they have such.
Natural weapons aren't light
Rules as written, there is no mention of the natural weapons being light, and hence they're ineligible for two weapon fighting. Whether the weapons are supposed to be light or not is hard to interpret, as Monster Manual commonly lacks the miscellaneous traits of monster attacks. However, it should be noted that the normal unarmed strike, also a natural weapon in a way, is similarly not light. Natural weapons and unarmed strikes are mostly unrelated (see the next paragraph) but it's still clear that "but it's just my hand" is not enough to qualify as "light".
Two weapon fighting requires wielding two weapons
Even with Dual Wielder feat (Player's Handbook, page 165), the rules require one to be "wielding" these weapons, a term not well defined for natural weapons. Jeremy Crawford tweets here and here confirm that natural weapons are weapons, usually not unarmed strikes, but still leave it unclear whether one can be considered to be wielding them. It's up to the GM's judgment, but I would personally disallow it for balance, because many of the lycanthrope attacks deal damage beyond normal melee weapon attacks (eg. 2d10 base for Werebear's bite).