I'm directly copying kviiri's answer to what is functionally a duplicate question. This question deals with the subject more cleanly and directly than the other one, however.
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. 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's unofficial 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.
I personally believe that Kviiri's answer is correct but that it would not be game breaking to allow natural weapons or unarmed strikes to be used with two-weapon fighting in most cases. This would be a homebrew rule, however, and something that needs to be cleared with a DM first. -Rykara