I mean, I was going for your cells being replaced one by one until you become the Thing. But, positing that you’re still you, but not you. Just like you are a Ship of Theseus even now, you aren’t the same You that you were or will be later.

Its a stretch, but is an interesting thought at least.

Edit: the interesting thing here is the Thing acted and behaved like the person, having absorbed who they were, fully capable of being them. They were the person, the person was still there, but they were more too.