The red and green lines you see are the world X, Y axes. The local axes go through the little orange dot marking the object's origin. Move your object in Edit Mode and everything will become clear:
You can add an empty in the middle and use it in the Mirror modifier to get the intended result:
Sadly this can't be done with curve's builtin bevel function yet, Mirror modifier on bezier curves acts on the generated mesh level, not directly on bezier data unfortunately.
You'll have to fake it with modifier equivalents.
Remove any native bevels you bezier curve has
Add a Mirror modifier with merge enabled
Add a Solidify modifier, set the desired ...
1. Why don't use Mirror modifier?
Mirror has an option "Mirror object". You can set here a central part of your plane or add an empty:
2. Linking object.
If you want to keep left a right wing separate, maybe it is a good option.
Copy one part of the ring with linking (Alt+D) then mirror it using Ctrl + M
Two objects will be separated, but geometry ...
According to the blender user guide, the mesh->mirror option found in edit mode is "exactly equivalent to scaling vertices by -1 around one chosen pivot point and in the direction of one chosen axis, only it is faster/handier."
I see from your image that you have the transform mode set to median point of selected, so any mirroring will be done ...
I can't see your mirror configuration as the object is selected in the right hand pane rather than the modifier tab.
You are also in edit mode which can affect mirror display.
In order to explore the mirror modifier, open a new general where you have a cube at 0,0,0 co-ordinates. The "origin" of the cube is in the centre and mirror uses the origin ...
Your mesh is fine. But the bone orientation is wrong. All bones stick out of the mesh but they should be inside. And there are too many bones for the thumb. You need to fix this first or you have a hard time animating this.
Your bone layout:
Good bone layout (Rigify meta rig):
Renaming the Bones
When you fixed this, give all bones the .R suffix ...