How do we define an equivalent when we are not talking about any particular reaction? Like, we say that equivalent volume of Oxygen is 5.6L, so we have taken the n-factor to be 4, and in CO2 we take n-factor = 2. What is the defining factor for an equivalent here?
The meaning of the word "equivalent" is relative to something. Fortunately, as a chemical term, that is still true. An equivalent is more completely called "a reaction equivalent" and is relative to the reaction. There are a few standard reactions (which are typically industry or discipline based) and it might be used usefully in limited (well constrained) situations. The problem is, especially when dealing with reactive gasses, that the words "uncontrolled" and "detonation" might also occur when explaining that you chose the wrong standard - the correct standard reaction is NOT always obvious or determinable in context. So, my best advice is that when someone uses "equivalent" ask what reaction, specifically, they are referring to. And when using the word yourself, always note the context. Examples: NCO equivalents, H(0) equivalents, H(+) equivalents may all refer to the same compound but would have different values.
You do not need to define the volume equivalent of a gas in relation to a reaction: you define it in relation to the number of molecules in the gas (or, more conveniently, the number of moles which is a convenient unit to measure the number of molecules).
Since ideal gases occupy the same volume if they have the same number of moles of their constituent molecules, volume (at constant temperature and pressure) is a proxy for the number of moles.