Could it be mod related, the display of lower damage?
I'm guessing that could be possible, depends on which mods you're using.
Or is it a normal game thing, if my skill goes up, so too does the damage displayed on the weapon?
IIRC that's how it's done in Morrowind -- if you find a new sword, the damage figures are the damage you'd be able to do, based on your related attributes and skills.
However, UESP lists a Daedric war axe as doing 22 damage, even though they specify that the figures are used
just for the site, and not based on what's actually seen in the game. However, 22 damage sounds really familiar for all of my characters (whether they could use the axe or not), so I'm guessing that the damage amount is just the weapon's base damage, not how much damage you'd be able to do with it. If that's the case, then seeing a lesser amount would probably be due to a mod.
I'm not at all positive though, just been wondering the same thing since you brought it up.