Can the 5D and the 5D iii share batteries?
No it won't, 5d3 it has the same batteries as the 5d2.
You can see the difference on these images.
Is that an older LP-E6? The recent ones have a date imprinted in a silver rectangle on the blue label. The reason I'm asking is that I acquired one like the one above and it doesn't communicate correctly in my 6d.