I don't buy that.

Owen is arguing that Artoo is now their property so all that junk about belonging to Obi-Wan would be exactly like getting a second hand laptop with an old user account still on it (or a second hand mobile phone with an address book of the previous owner's numbers).

From an owner's point of view.

Yes there is more to it but we don't know exactly what yet.

From an audience point of view it's that information that makes him a 'person', and a witness to the earlier part of the story.

If Owen wiped that he wouldn't have his mission from the Princess in his mind or possibly his past memory of being 'friends' with Threepio.

As an audience we have an emotional investment in that 'data' like we would with Luke's memory of his Uncle and Aunt and Ben after they died.