EDIT: I've changed the answer a few times because I realised you can achieve different results easier than with my first solution. Here are some examples from very simple to a little more complicated to get more control.
2. EDIT: Since you've added in the question that the wings should be animated and rotating around an axis, I've made new version which is a bit more complicated, you'll find it at the end of this answer together with a blend file.
But here is the most simple version for mirrored procedural textures without the need of unwrapping UVs (let's say you want a 3D texture not mapped onto the surface): Use Texture Corrdinate > Object with a Vector Math node set to Absolute, this way the texture is mirrored on all axes across the object's origin.
If you want to mirror only on one axis, you can simply do this by using the Separate XYZ node and only setting that value to Absolute which axis you want to mirror, for example X, then Combine XYZ to get the vector for the texure:
If you want a mirrored texture but somehow make slight variations between the left and the right side, you can do this by using the same Voronoi texture twice and mixing the left and right side together using a Greater Than > 0 node and a Color > Mix RGB node, for example in the next image thetexture is mirrored, but one side is greyscale and the other colored:
A last example for something you simply couldn't do if you would have to use UVs: here you have mirrored 3D volumetric balls inside the object.
So, here comes the version to be moved around (more or less freely):
To achieve that, first of all I need the wings object. It's the right wingside mirrored on an empty's X axis (cross shape in example) and parented to it.
To rotate the left wingside correctly when the right is rotated, I'll take a second empty (cube shape) with a Copy Rotation constraint with the wings object as Target, copying and inverting all axes and setting Target and Owner to "Local Space".
This cube empty is parented to another empty (circle shape) which is used to moving and rotating the wings around in world space. The circle empty has Copy Location and Copy Rotation constraints, both targeted to the circle, all set to "World Space" except for the rotation Owner > "Local Space".
All objects have their origins in the same location. To combine them in the Voronoi material I use the follwoing node setup:
With this setup you can move and rotate the complete wings by the circle empty and flap the wings by rotating the wing object:
Because this is all very complicated, here's the file: