ELI5: Birthday paradox vs Gamblers fallacy

Hi all,

Im really stuck in the way I think the brithday paradox and the gamblers fallacy contradict one another and i cant find a awnser that explains that they dont contradict (that i can follow with my logic).

What I think is that the paradox proves that probability grows exponentially when stacking, but the fallacy proves that it does not. Is this comparrison a paradox in itself or do I miss something?