Even stormcloaks say by the eight.

Really? Ouch.

I've noticed a scattered few characters will refer to the Nine Divines rather than Eight. Erik the Slayer is another one. Whenever we stumble across some epic scene in a dungeon he'll almost always blurt out, "By the Nine!"
Kinda makes sense in his case. Rural farming communities (like Rorikstead) have other, more immediate concerns besides some treaties made hundreds of miles away by distant governments, and the Thalmor Inquisition's more focused on the larger cities for now besides. There's probably more than a few people in those villages who still openly worship Talos.
(Plus Erik's kind of a Tiber Septim fan. And I've caught him shouting "Skyrim belongs to the Nords!" when charging at non-Nord bandits sometimes. Guess he's pro-Stormcloak?

)
I figure it has to be deliberate on Bethesda's parts rather than an oversight. I mean, why bother writing the lines, having VAs record the lines, and then putting them in the game only for certain characters otherwise?