The confusing thing is that it seems to store the uv info in the skin? Since it is a rigged model.
I think this may be vital info in solving the issue
No, as far as I know there is no way for UV info to be stored in the skin modifier. The UVW Unwrap modifier should have been applied before the skin modifier and then collapsed. Any time you place the Unwrap UVW modifier on the mesh, it will reveal the UVW map. You simply do not have it in the modifier stack right now from your picture.
To figure out what happened I suggest the following:
1. Make a copy of your 3ds max file and work on that.
2. Pplace the Unwrap modifier right above the mesh and open the editor to see the UVW map. If it looks good with an orderly layout of UVs and no overlapping UV verts/faces then close the editor.
3. Turn off the skin modifier in the stack by clicking the eyeball next to it.
4. Select the mesh and export it via .fbx using Export selected.
5. Import the .fbx mesh into blender and see if this time the model texture seems to be placed properly. If not, check the Blender UVs to see if they look the same as what you saw in Max.