You cannot do what you are trying unless you parent an object inside another. If you create an object and rotate it. Then you take another object and make it the child of the rotated object and then freeze transforms on the child. The childs LOCAL axis will now match the parents rotation angles but will have a rotation vector of <0 0 0> relative to its container.

Think of it this way. When you freeze transforms the object pivot will align with the pivot of the container it is in. In the case of an object in world space that would be the world axis rotation.

Now create a box and rotate it say <10,20,15> and now create a second box. look at it's local axis. It matches the world axis and it's rotation vector is <0 0 0>.

Now parent make this cube the child of the rotated one (1st pick the new cube, then shift select the rotated one, and press p).

Now the new cube is a child of the parent and it's local rotation vector still matched the world axis and is <0 0 0>.

Now select the child (the new cube) and freeze transforms. The local axis of the child will rotate to match the parent cube <10 20 15> but the local rotation vector for the new cube will be <0 0 0>.

Now remove the new cube from the parent. It's rotation vector stays the same orientation but becomes <10 20 15> relative to it's container - WORLD SPACE.