Whenever I have "Executioner Axe" equipped my damage remains the same with any amount of rage stacks.
Using it as 2h, 1h with shield or dual wielding it with another weapon ("Obsidian Edge") does not make a difference. ( I have not obtained 2 "Executioner Axes" to test out dual wielding those despite the "Unique" tag)
Switching to another weapon ("Obsidian Edge") without wielding "Executioner Axe" lets me benefit from rage again aka my damage visibly changes.
(Once ill figure out how to make screenshots on my steam deck ill try to post some)