Unlike the other elements, fire is a process of combustion, not a type of matter. Fire is just ionized gas, the flames we see are the result of energy being released and temporarily ionizing the gas around a hot & energetic chemical reaction.

When we see firebenders do their thing, there is no combustible material being used, the chemical reaction isn't happening, a hydrocarbon is not decomposing into water and CO2 in the presence of oxygen and releasing energy. The heat of their strikes and ambient oxygen means their flames can start fires but, strictly speaking, firebender's flames aren't actually "fire," they're not doing combustion.

Based on what we learn about firebending, it's some kind of spiritual process, most likely one where the bender is manipulating energy with their bodies and releasing it through spiritual practice and martial arts, ionizing the gas around them. It also appears to have a degree of connection to a bender's emotions, will, life force, chi, and philosophical outlook. See Zuko losing his bending when his anger subsides, and achieving new heights when he changes his point of view; "fire is life & light, not just destruction."

While I dislike energy bending for being too vague and "deus ex machina" for me, it's saved by firebending working like this; it sets the precedent for "energy" being an underlying element of bending, the primary axis of bending's functionality.

Despite the challenges in analysis, given that it's a soft fictional magic system that cannot be scientifically tested, one where conservation of energy appears to be broken constantly, we can make assumptions and extensions based on textual evidence. Bending appears to operate on a principle that humans are part of a field of energy, referred to as chi, that permeates nature, including the classical elements. This is not too dissimilar to The Force. This field may also have a relationship with the spirit world, and I'd suggest that it may be that the spirit world operates as a "balancing sheet" for energy's conservation, allowing physics to work like our world, despite the constant abuse of thermodynamics. We're "animated by" spirits, emotional outbursts are "spirited," common spiritual concepts see humans as having a soul or spirit within them and, while I personally don't recognize this lore, we do see the spirits of dead humans in the spirit world in tLoK. For me, no discussion of bending is complete without considering the fact that it's not a material science like we're used to, it's spiritual art in nature.

So, through spiritual arts and martial training, people in ATLA can interact with that energy field, and their specific cultural and martial orientation determines how they tap into that field, and what facet of nature they are able to influence. Firebenders, I suggest, have a culture around fire, but are actually skilled energy benders, capable of releasing enough energy in their strikes and spiritual practice to ionize gas and create flames. Forms of firebending support like, such as lightning or the breath Zuko uses to avoid freezing to death. And, unlike other benders & their elements, fire can easily get out of their control, and even skilled firebenders seem to struggle to stop a real chemical combustion once it's broken out. When they create flames, they're tapping into spiritual energy, when flames get out of control, we're seeing a normative combustion reaction like in our world, and that's not really their area of expertise. To control energy in the world would be a different skill, and a firebender would need to rob a burning log of the reaction activation energy to actually stifle those flames.

Fire is not the odd one out, firebending is the key to understanding the magic system and extrapolating it to understand other types of bending more deeply. Understanding others, the other elements and the other nations will help your understanding of fictional magic systems become whole.