What would be the impact if I ruled that the Lizardfolk's Natural Armor stacked with Barbarian's Unarmored Defense?
It would make Lizardfolk Barbarian an exceptionally strong combination, compared to other similar choices for Barbarian race. In the 5E design of bounded accuracy, a +3 passive increase to AC is a huge deal.
A player character who took this option and tried to optimise their barbarian for max AC might cause headaches for a DM, where the barbarian character could handle threats that the rest of the team were not a match for, all else being equal.
It is unlikely to break the game as a single change, the barbarian can still be hit by monsters, especially in higher level play. And they may outshine their colleagues in battle more often, but it probably won't be 100% of the time. However, I cannot think of a good reason to allow it.
If the player's concern is that they have lost out on benefits because they double up andconceptually, but don't stack and provide any improvemnt, then potentially you could allow a boon or feat to grant a smaller increase for the stacking. E.g. perhaps a Feat that required natural armour and granted a flat +1 to AC, regardless of whether that natural armour was being used in the current AC calculation.