There are other chemical reactions that produce heat, but fire is specifically a rapid oxidation. There are other oxidizers though that can result in an oxidation type reaction though, a few are even better than oxygen at it even, like fluorine (which is why pure fluorine is such dangerous stuff)
Chlorine Trifluoride is a better oxidizer than Oxygen itself. Hell, the germans in WW2 tried making self igniting flamethrowers with the stuff. But stopped because it was too nasty to work with.
At that point, they might as well just spray it directly on people. Screw using a fuel to ignite when the trifluoride is enough to set fire to literally anything. Even wet sand can be burnt by it.
Which reminds me of this quote "It is, of course, extremely toxic, but that’s the least of the problem. It is hypergolic with every known fuel, and so rapidly hypergolic that no ignition delay has ever been measured. It is also hypergolic with such things as cloth, wood, and test engineers, not to mention asbestos, sand, and water-with which it reacts explosively. It can be kept in some of the ordinary structural metals-steel, copper, aluminium, etc.-because of the formation of a thin film of insoluble metal fluoride which protects the bulk of the metal, just as the invisible coat of oxide on aluminium keeps it from burning up in the atmosphere. If, however, this coat is melted or scrubbed off, and has no chance to reform, the operator is confronted with the problem of coping with a metal-fluorine fire. For dealing with this situation, I have always recommended a good pair of running shoes. "
There can be a release of energy without oxygen, but it will not result in a fire, just an explosion. You can make arguments for things like the sun "burning" in the vacuum of space, but that is fusion, and a far different thing than lighting a match.
It blows my mind, really. My own intuitive understanding of gravity just doesn't extend to that kind of reality (he says with a broken collar bone due to essentially 'too much gravity')
There are plenty of things that can cause fires that are not oxygen, and don't contain oxygen.
The halogens, Fluorine and Chlorine in particular, are powerful oxidizing agents on their own and can produce flames in the same manner as common flames.
Here's a report on the spectra of flames produced by combustion in a Fluorine atmosphere (PDF warning).
Some metals (the ones that require super special fire extinguishers) can keep an exothermic reaction going (magnesium I believe, and sodium I'm pretty sure) but rapid oxidation (rusting) is the most common method of combustion.
Oxygen is so combustible that it's toxic to life, and would have killed life as it was a billion years ago (number from my ass) which produced oxygen as a byproduct of photosynthesis if it weren't for the development of mitochondria.
A few other things (elements) can burn without oxygen, but not many, and they normally need a pretty large activation temperature.
So is it just the definition of "fire" itself? Like only something burning with oxygen is "fire" but if it's another fuel source it's not technically "fire" but we call it fire anyway?
Burning with oxygen is oxidation, things oxidize when the element oxygen binds with another element, which it's very prone to doing. Oxidation with iron is rust, it just happens much more slowly (but still exothermically) than when biological compound oxidize they release energy more rapidly, rapidly enough to cause other nearby organic bonds to break and expose themselves to sites for more oxidation to take place.
There can be electrical fires, chemical fires, classic fires, and self fueling fires (the kind you were originally asking about). There are probably more categories now. Always be sure to use the proper fire extinguisher for the fire at hand.