What we have confirmed so far is that they were identical copies at the moment of the great sundering, but have been on very different - but simultaneous - paths ever since. Theoretically, the different balances of Light/Dark, the Source getting rocked by 7 Calamities, and the butterfly effect should ensure the worlds have little in common by today. (Theoretically.) The First was shown to be different in the way Arbert and Co. had similar, but different, job titles and abilities. (Their names and gear were borrowed from Eorzeans so they'd blend in, though.)
As far as we've seen proof for? If the soul has the Echo and can transcend the flesh and death, they could force themselves into another living person's body, extinguish them, and warp their body to look like their old one. But that seems like a terrible thing to do. If you had clones of yourself it ought to be pretty easy. (Solus - perhaps actually Emet-selch, perhaps not - actually thanks Varis for conjuring up all those clones and saving him the trouble of warping other bodies to look like him.) If you had a powerful technologist like Amon or Aulus maybe you wouldn't need the Echo? Maybe? And then there's whatever the Warriors of Darkness / Unukalhai are doing, I guess. They all died and then showed here with physical forms. That was ... just never addressed.
I think Y'shtola just used a risky teleport spell and got "stuck", no? Her body/soul should still have been entwined, even in aetherial form. I think it's being awash in the great torrent that is the Lifestream for too long is what damaged her.