Does the Lizardfolk's bite ability stack with the Unarmed Fighting Style (from Tasha's)? What I mean is if I'm a Lizardfolk Fighter with the Unarmed Fighting Style, can I make an unarmed strike with my mouth that does 1d8+STR piercing damage?

Edit: clarity

Well, the Lizard's Bite attack is defined as a Natural Weapon, but the race description explicitly says that it can be used to make Unarmed Strikes. The Unarmed Fighting Style is perfectly compatible with that... although to get the full benefits you still need to forego carrying a weapon or shield... even though you're not actually using your hands to make the attack, the RAW still states that simply wielding a weapon or shield reduces the bonus down to a d6, which is a Lizardfolk's normal damage anyway.

Sorry, should have put I'm not going to use weapons or shields! Thanks for the confirmation! Now, I can play my Lizardfolk Rune Knight that uses Giant's Might to turn into a T. rex and bite people.

You can still hold other stuff, interestingly. Like a lantern or potion or the gnawed-on limb of your last victim opponent.

Well, as a rune knight, I can hold something with the Fire Rune on it, which allows me to shackle a target in fiery chains after making a weapon attack (bite is technically a "natural weapon"). 

Reptilian? Check. Humongous size? Check. Fire Breath? Check. Let me get some wings, and I'm a bona fide Dragon!