To me, these comments are what's interesting to consider about the Fighter.
What about it is unique?
A quick scan of it's class features (or the other 97 pages of this thread) answers that.
Fighting Style isn't unique.
Second Wind isn't entirely unique (as self-healing is built into the Way of the Open Hand Monk's Wholeness of Body & the Paladin's Lay on Hands).
Extra Attack isn't entirely unique (as the Fighter just gets more at higher levels compared to Barbarian, Ranger, Paladin, and several other subclasses).
Indomitable also isn't entirely unique (when compared to the Paladin's Aura of Protection).
Having more feats is a cool feature, but it doesn't provide unique identity.
So what does that leave? Action Surge.
That was my point – that the entirety of the 5e Fighter's unique identity really rests on Action Surge.
What stories do its subclasses tell?
As many as your imagination can supply. Part of this is your backstory, the rest happens during play. So I don't really see how the book alone could ever answer your question.
You're misinterpreting me, or I was being unclear. By "stories" I meant unique identity. I was agreeing with Mike Mearls' sentiment on the TomeShow that the Fighter's subclasses lack the unique identity that other classes' sub-classes possess.
For example, a Rogue (Thief) breaks in and steals, while a Wizard (Diviner) foresees the future. And a Fighter (Champion)...uh... Well, a Fighter (Battle Master)...he...hmmm...
What does the fighter do when not swinging a blade or shooting a bow?
Well that's a bit vague. Are you talking about within an encounter? While exploring? While in some social situation? Overall, in the big picture, during downtime? What?
In general I guess my answer would default to skill/tool checks. Wich is what everybody else does when not fighting....
If this is related to the stories each sub-class tells then that's up to the playerto determine.
If you compare class features of the Fighter to, for example, the Ranger...you'll see the Ranger has several more class features which pertain to non-combat situations. And it's not just the Ranger. The Barbarian and Paladin also have more non-combat oriented class features than the Fighter.
Now, when I bring this up, people often say "well, it's called the Fighter for a reason, dummy, it's supposed to JUST FIGHT." And my response to that is: "Oh good grief, this is going nowhere."