Yes, in a fantasy setting, one can -- and many do -- say things like "the elves of the Forest Conclave all learn the art of archery". But real cultures seem rarely to be like that, and even in fantasy, probably not all cultures are like that.
And there's your problem. You're using the world 'culture' in two different ways, and wondering why they're not the same.
'Culture', in terms of character generation, is a word that allows for a collection of general bonuses that are not strictly related to either the character's race/genetics/physical body, or to the character's personal choice of pre-class activities (background/apprenticeship/job/etc). It has a broad match with the idea of 'culture' as might be used to describe nations (eg: France) or peoples (eg: Kurds), but it is not the same thing.
It is a game mechanic term, in the same way that Intelligence is a game mechanic term that doesn't necessarily match up with IQ tests and such. It's close enough to the real-life meaning that the casual player can easily grasp the intent behind what goes in that box, but it is not a rigorous scholastic term.
Personally, I would make use of the game mechanic Culture as describing a broadly similar collective behavior set that covers baseline social interaction methods. A merchant culture, or a nomadic culture, or a warmongering culture, or a fishing town culture, etc. Each of those do not have to be the entirety of a culture. Waterdeep isn't solely and entirely composed of a merchant culture, but that merchant culture is definitely a part of Waterdeep. Not all orc tribes are warmongers, but a warmongering culture definitely exists within at least some orc tribes.
So you could be an elf from Valenwood, but Valenwood isn't your culture; it's just where you grew up. Rather, you might have been raised in a culture of elitist nobility, that just happens to have been in Valenwood instead of some other part of the world.
This
does bleed into the current 5E Backgrounds a bit, because Backgrounds are kinda fluid in what they represent. An Acolyte is very different than a Soldier, which is different from a Noble. The Noble background is closer to what I described above as a Culture, but a Soldier is a profession chosen after you grew up (usually). An Acolyte is sort of in between, where is might be chosen after adulthood, or it might be something where you were raised within a religious order.
So Backgrounds cover both too many and not enough things. Separating things out a bit would allow better-defined characters.
