Whenever we construct a set, we do so in such a manner that always we have a certain property or a set of properties that the members of the set specify. Even when we create an arbitrary set of alphanumeric elements, there's still some basis or property in our mind that we have chosen while constructing a set that is fulfilled by the elements of the set. Is this statement of mine in any way similar to or analogous to axiom of specification?

I have comprehended the idea of axiom of extensionality by the following example, lets just say that I am sitting in a closed room with a sound source. Now there's one way in which we might get the same loudness level such that we here the sound wave directly, right. The other way is to block the direct sound waves and position the listener in such a way that the the combined wave at my ears at that position due to reflected waves gives me the same loudness as earlier. But if I am unaware of the way in this loudness is achieved, I might confuse the system to be just the same, right? And hence the axiom of extensionality which is based majorly on quantifications rather than the way in which an event would yield the same output, right?

Is this the right way to understand the way in which sets are constructed and what makes sets equal?