Hi @fredi
Many thanks for the post. Ret assured, this isn't the first time that the mismatch between the world axes labelling and Move command axes labelling has caused confusion 🙂
The basic logic is: if you use the Move command and pick an object in the canvas, we use the face you picked as reference for the manipulator axis directions. In most cases, this will be based on the normal direction of the face you picked: 'Y' will be normal to the selected face, and 'X' & 'Z' will be orthogonal to that. There are some special cases (e.g. picking a cylindrical face will align the manipiulator with the centreline of that cylinder, with 'Y' being the centreline direction).
So the directions of these axes will be purely based on the selected face; they won't necessarily correspond to world space.
If you wish to move an object and keep the X/Y/Z axes aligned to world space, the easiest way to do this is to pick the object from the browser rather than the canvas:

If you to do this, we default to aligning the manipulator with the world* coordinate system. This should give you the behaviour you are looking for.
(*This is assuming you're not yet working with components & subcomponents. When you start creating Components: each Component has its own coordinate space, and transforming a component will transform its coordinate space. So if you transform a component, any objects within that component will align to the parent component's coordinate space, not world space)
There's also a 'Set Pivot' option inside the Move command, once you've picked an object to move. If you click this, you can pick any point to reorient/relocate the axes. If you pick the world Origin point from the browser, then click the tick icon to complete the Set Pivot, this should also align your manipulator to the world axis directions.
Does this help clarify the behaviour? If there's still any confusion, please do let us know.
Thanks,
Jake

Jake Fowler
Principal Experience Designer
Fusion 360
Autodesk