Lightning is linked with firebending through many reasons.
But first you have to think what is fire and lightning:
Fire -– a state, process, or instance of combustion in which fuel or other material is ignited and combined with oxygen, giving off light (same as the glow of lightning), heat, and flame.
Lightning -– a brilliant electric (creates light) spark (fire) discharge in the atmosphere, occurring within a thundercloud, between clouds, or between a cloud and the ground.
Both fire and lightning are linked due to the flow of energy. Fire, when has no fuel, ceases to exist. While at the same time, lightning appears for seconds.
In the Indian culture fire was called AgniAgni, and it had has three forms: fire, lightning and the sun.
In Avatar the Last Air Bender, lightning was considered "Cold-Blooded Fire". And in several occasions (nonot just in the show), the lightning creates fire when it stuck the surface of a flammable object, letslet's say wood.
Also, Lightning and Fire burns.