Can I add a object to the animation without to animate it so it just moves with the other transform or is this not a thing in blender?
I want to add this axe to my player model and to make it so it moves with the characters animation.
You can give your axe a Child Of constraint with the hand bone as Target, or select your axe, shift select the armature, switch it to Pose mode, and press CtrlP (Parent) > Bone.