I see how this might be confusing so here you go:
If you own dwarfs you get a little DLC flag on your account that unlocks the Axe for you. No achievement is needed. Thank you for owning Dwarfs!
If you get the Not a war hammer achievement in Dwarfs Free to Play the next time you get into a KF server of play solo you'll get the Gimli that Axe achievement, unlocking the Axe for you. Thank you for playing Dwarfs Free to Play!
If you own the full version of Dwarfs and unlock Not a war hammer it doesn't give you Gimli that Axe because you don't need it to unlock the Axe. You already unlocked it thanks to owning Dwarfs. If you're a completionist and want the achievement in KF simply get the achievement in Dwarfs Free to Play. Free to play and the full version have their own achievements.
Hope that clears things up.