As far as I understand it, archetypes are more of Pf2e's way to do 'multiclassing', where you choose feats from an archetype instead of your normal class feats.

So you're right, there is no direct correspondent to DnDs subclasses in PF2e. But many classes still have subclass-like specializations. They have different names for each class, for example 'ways' for the gunslinger, 'instincts' for the barbarian or 'arcane theses' for the wizard. These in themselves tend to have a little less meat than 5e subclasses, because you still have a lot more customization options with your feats. Mostly they modify a core element of the class and also give acess to specific class feats.

Sadly, the fighter seems to be be the odd one out and is designed to not have any such specializations. Maybe so it can be a beginner friendly class that doesn't overwhelm the player with choices..?

So as I see it you might:

a. just make some appropriate fighter feats for your culture that any fighter can choose (with whatever restrictions you place on that as the GM, regarding backstory, etc.). This would be the easiest in my eyes, as you could orient yourself on existing feats for how much these should be able to do.

b. you could make this an ancestry/heritage thing, seeing that it should be tied to a culture. Maybe not exactly what you want, if it is supposed to be for fighters only. Could work as just a generally available ancestry feat/series of feats that players can choose. However these often don't make mayor changes to a character and rather give a small bonus to something.

c. make a new class that gets all the same stuff as a fighter, except for some things you replace with your own homebrew. This might get a bit more complicated than creating additional feats.

d. figure out how to modify the entire fighter class to offer a subclass-like choice. I wouldn't know where to start on this one. This would easily break something in how the game is balanced and feels like a job for people who are on a level of understanding this game on par with the designers at paizo.

Just as a final note: I haven't really started to homebrew much in PF2e myself, so my ideas on this might not be super accurate.