Nevermind... I solved it (after hours of testing):
In MP the fdf_axe only works if the unit holding the axe is NOT group leader. As soon as a unit equiped with axe becomes group leader, the axe doesn't work for him any more.
If he then joins another group, it works again. Weird but true.
Now, can anyone explain why the fdf_axe behaves this way? I'm puzzled!!!