>>358262479
Try the Mesh Data Transfer add-on, it's free:
https://mmemoli.gumroad.com/l/tOKEh
Here's a 3 minute tutorial from the author Maurizio Memoli:
https://www.youtube.com/watch?v=CvMeK_IIw-Y

I was able to use it to transfer the shapekeys from my original model to a subdivided version (for testing) because the UVs stayed exactly the same. (It only looks different due to the camera angle and the subdivision smoothing things out.)

It has more options to sample from as well, maybe one of the modes will be suitable for your case. 
(Image related:) Trying out "Local" with "Closest" or "Active UV" with "Closest" might help in your case, considering that you said the meshes are identical, though it could work even with small changes.

IIRC, modifying the mesh changes the vertex order, which is important for Blender to know how to modify the Basis to be like the blendshape, but this add-on tries to guess which vertex on the reference model corresponds to which vertex (or even vertices) on the modified model. So, normally even modifying one vertex destroys the shapekeys because the entire vertex order gets changed, which, by the way, is also why Blender doesn't allow you to apply mesh-modifying modifiers once there are shapekeys present.

2.93 seems to have recognized the worth of being able to transfer mesh data like this, which is why it's possible to transfer mesh data natively as well now, apparently (but I haven't looked into that yet):
https://docs.blender.org/manual/en/latest/scene_layout/object/editing/relations/transfer_mesh_data.html