While Onyx' answer in that the bones use their own space independent of the world space; there's a bit more complexity to that:
Out of the box, Blender's bones are in Yup coordinates system, and Blender uses the bones' local axes, but taking into account whatever rotation you have in the rest pose, and independently of the rotation in pose mode.
Let's take a simple default bone for example. If we enable the armature object's Axes in the Viewport Display panel, we see this:
In this case, if we use the default worldspace gizmo to move "up", which is +Z, we would see the bone's Y location getting change. Because the worldspace +Z is his +Y.
Now, if you rotate that bone -90 on its X axis, you obtain this:
At this point, moving the bone up would still change its +Y value. But if you hit ⎈ CtrlA → Apply Pose as rest pose, now moving the bone up will change his +Z:
In other words, if you want your bones to use any coordinates' system, you need to set it up before animating, either while making your rig in edit mode (recommended) or from the pose mode by applying the pose as your rest pose (which WILL break any animation already done, and might also break your rig depending on how it is made).