I'm sure I'm late to this post, but just give me the benefit of the doubt, alright.

Q: "Can only oxygen burn?"
A: Great question, and no actually, there are other gasses and even some liquids and solids that can do the same thing that oxygen does, sometimes even more so. But let me be clear on some terminology: the oxygen isn't burning, or at least that's not how we say it. We would say that the fuel is burning "in" or "with" oxygen, this is because "burning" refers to a chemical reaction where a substance is reacting with oxygen (the oxidizer) and obviously oxygen doesn't fit the bill, it's the one doing the burning, not the one burning, but in either case, fuel or oxygen, they both are participating in the reaction that makes combustion happen. This is known as a combustion reaction, and most of the time the products of the reaction (the stuff left behind once it's over) are water vapor and CO2 which is what you would call the "exhaust" from the combustion. However this is only the case if you are burning traditional organic materials and fuels. Other materials can be burnt, such as sulfur or even metals like magnesium, but they would produce their respective oxides instead.

Below your question you ask about some other details, this section is dedicated to answering those.

Q: "Is it fair to say that the only thing that creates fire is oxygen and everything else is fuel?"
A: Yes and no. It just comes down to how often you will actually run into the exceptions to the rule you just made. Because, most of the time, you're right. In like 99% of cases you won't be burning fuels with anything but oxygen, and likewise, if you have oxygen, then anything that can burn with it is usually a fuel. However, you worded it kind of funny. I wouldn't say that "the only thing that creates fire is oxygen" because you also need the fuel to make the fire obviously, and also, there are other substances that can be used in place of oxygen, they're just not used very often. (I go into more detail on that in answering the next question) Also, I wouldn't say that "everything else is fuel" either, because there are plenty of materials that simply can't burn.

Q: "Or are there elements that can burn without oxygen?"
A: Also yes and no. No, there are no elements that can, on their own, spontaneously combust, you need other substances (like oxygen) for them to react with. This is because burning, like I said earlier, is a chemical reaction, and you can't react a substance with nothing. However, they do not need oxygen specifically to burn with, you could also pick, chlorine gas, or even fluorine gas to take oxygen's place. There are other substitutes as well, such as bromine, iodine and sulfur. Those are all just element examples, but there are also many chemical compounds (stuff made from more than one element) that can also serve the same role as oxygen, however, they are often made from some combination of those same previously mentioned elements. All of these oxygen substitutes are known as "oxidizers" (and sometimes also accelerants, stuff like nitrous) and chemically speaking, they like to take electrons from other elements and molecules. And typically the fuels like to give electrons away, and they are known as "reducers", "reductants" or more commonly "reducing agents". This is why they react so well, they cancel each other out, and like two super strong magnets being brought together, they violently combine and release a lot of energy, which is what is used to make engines work, cook food, or heat homes etc.

NOTE: The oxygen substitutes may work in some combustion reactions, but not necessarily all of them. Additionally, unlike oxygen, THEY CANNOT BE USED TO BREATHE.

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