If we assume a planet the size of Earth with an equally dense atmosphere of nitrogen and methane, rather than oxygen, and it's in the goldilocks zone. Would a hypothetical alien species be able to use oxygen for lanterns and stoves the same way we use methane (or natural gas, etc)?
If so, what primary differences in ratios and/or densities of gasses would have to exist? If not, why not?
Yes, it would work.
The difference between "fuel" and "oxidizer" is rather made for convenience, but it's not a fundamental distinction at the macroscopic level. The only differences are at the microscopic level, related to what the electrons do during the reaction. All such processes are redox reactions (reduction-oxidation), where one participant is the reducer, and the other is the oxidizer. That simply means the reducing agent loses electrons, while the oxidizing agent gains electrons; once the electron balance is final, the two components are bound together and the reaction is complete.
The oxidizer doesn't even have to be oxygen, it can be many elements or compounds hungry for electrons.
The bottom line is - you can burn either gas in an atmosphere made of the other. The gases don't care, as long as they react with each other. There are videos online where you can see chemists burning oxygen in an atmosphere of hydrogen, or burning hydrogen in an atmosphere of chlorine, etc. All these are redox reactions.
As to what's the minimum percentage of methane in the atmosphere that could sustain an oxygen flame, I am unsure. But it must be similar to the situation on Earth, where a low-enough oxygen contents in the air will eventually extinguish any flame. I would guess that 20% methane would work just fine for burning oxygen, but that's just a guess, so take it for what it's worth.
As others have pointed out, a methane atmosphere would indicate a very different chemistry at the planetary level, which may or may not be able to sustain life in general. It would definitely not sustain life as we know it.
Also, the Goldilocks zone for a methane atmosphere is different from the same zone for an oxygen atmosphere - it's further away from the star, because methane retains infrared better than oxygen (methane is a "greenhouse gas").
Fire and Flame 19 - Oxygen Flame in Hydrogen (Royal Society of Chemistry)
Fire and Flame 21 - Oxygen Balloon in Hydrogen (Royal Society of Chemistry)
I agree with Cipher's answer, but just to add, from a certain point of view, Oxygen could be stored and used like a fuel in a situation like that.
Now, as far as aliens doing so, I have some doubt that you can have life (intelligent life if they're using fuel) on a planet with a Nitrogen/CH4 atmosphere cause I don't see how that would actually work and how the resperatory cycles would balance out. I looked into that a little bit and it gets pretty complicated, so, I don't want to say it's impossible, but I suspect it's unlikely, but, if we take your scenario and work with the assumption that there's intelligent life on a Goldilocks planet that breathes N2/CH4 (maybe throw some CO2 in there), on a rocky planet with oceans, I see no reason why they couldn't use Oxygen as fuel, at least, running "oxygen lines" inside their alien homes to make fire, perhaps to cook with - sure. It seems like it would be a useful convenience. You don't gain much energy pulling Oxygen from Silicate rock or Iron Oxide or from water and then using it to burn in a methane atmosphere, so I don't think it would be an energy source, unless they use bacteria to produce oxygen and then, maybe. Oxygen wouldn't concentrate underground the way hydrocarbons do, so in that sense, it wouldn't be drill-able, but as a transportable "fuel like" substance, it would work. I see no reason why that couldn't be done.
If we ever set up an astronaut colony on Titan, which, while much farther away from the earth than Mars, it has several advantages, one being, the surface pressure is manageable, so structures wouldn't need to be able to contain a significant fraction of 14.4 PSI. They'd need to be mostly air tight, but compared to the structural integrity needed to not lose atmosphere to a vacuum, that's easy.
Astronauts/colonists on Titan would be able to collect rocky CO2 and H2O from the Moon's surface and use that to grow plants inside and in time, they would probably have an abundance of oxygen and under certain circumstances, that Oxygen could be used in a similar way to fuel. I think the scenario the way you wrote it is very unlikely, but oxygen being used like fuel on a methane or hydrogen abundant atmosphere, more likely some kind of astronaut colony, there's no reason why that couldn't be done that I can think of.
PS - That's on the speculative side for an answer here, but that's my thoughts on the question.
I think in the spirit of the question, the answer is yes. In an atmosphere like hydrogen or methane that can react with oxygen, oxygen is a resource that allows combustion to occur and therefore represents an energy reserve. For example, here's a demonstration of an oxygen flame in a hydrogen atmosphere.
