A CLASSIC case! This is a major issue going back to ancient philosophy. The example used then was the ship of Theseus (a Greek hero). Imagine you have the ship of Theseus and a similar ship side by side. First you switch one part (the mast, say). Is the ship of Thesus still the same? Many of us want to say 'yes,' but then we get puzzled as more and more parts are switched until eventually it seems the ships have changed places. One route that philosophers have taken might bring peace to your family: some philosophers distinguish a strict sense of identity from an identity that is "popular and loose." On a strict view, you are right. Any object with parts is not the same if even a single part is removed. This is technically called mereological essentialism. According to mereological essentialism, your father's body today is not identical with the body he had as a boy. You might even suggest to him that while he went to first grade, that (pointing at his body) did not. You can retain mereological essentialism while also allowing that sometimes we can and should meaningfully speak of sameness of identity that is not strict. In the later, though, there will be different conventions that come into play. So perhaps there is a way to allow that you both may be right? If push comes to shove, however, I am on your side on this. For a defense of our view, see the book Person and Object by Roderick Chisholm.
'We can only live in this "here and now" moment . . . in fact , there is no way we can ever live out of it . . . is it not?'
I am not sure what is supposed to meant by living in the present instant ("moment" I think has more to do with action). Living at an instant seems as impossible as living at some other time, because there isn't even time to draw breath in an instant. In any case I do not believe that there is something called "the present instant", so I don't see how we could live in it (at it?)
It (the present instant) is an abstraction, and it is not, in reality! I do believe there are present times, though, such as the present day or hour. The trouble with the instant is that it is not a time.