Two Identical Meshes But Only One Works with Skeleton

### Tested versions

4.3-stable

### System information

Godot v4.3.stable.mono - Windows 10.0.19045 - Vulkan (Forward+) - dedicated NVIDIA GeForce GTX 1060 6GB (NVIDIA; 32.0.15.6094) - Intel(R) Core(TM) i5-8400 CPU @ 2.80GHz (6 Threads)

### Issue description

I have to absolutely identical meshes, both exported from Blender, one of them was exported with the Armature and the other was was exported alone.

I **already** have the skeleton for this mesh in my Godot scene, I only need to change between meshes, but for some reason it only works if I export the whole armature + mesh from blender and not only the mesh, which is all I need. It seems like the vertex groups are getting lost somewhere or not being loaded properly.

Exported w/Armature:

Exported only Mesh:

Quick video:

https://github.com/user-attachments/assets/fd84bac0-77a3-4baa-a030-147b67e45573

### Steps to reproduce

1. Create an Armature in blender and animate a simple mesh
2. Import the full model to Godot (mesh + armature) everything works fine
3. Go back to blender and create/modify the mesh (just the mesh)
4. Import only the mesh into godot
5. Go into the skeleton and change the mesh for the new onw
6. It doesnt work

### Minimal reproduction project (MRP)

[bug-report-meshes-anim.zip](https://github.com/user-attachments/files/17670795/bug-report-meshes-anim.zip)

Start by opening the attached bug-report-meshes-anim.zip and reproducing the steps with the Blender exports in Godot 4.3. Compare the imported mesh and skeleton data between the full armature export and the mesh-only export, then verify the replacement mesh under the existing Skeleton reproduces the reported failure.

Written by the indexing model from the issue text.