If you take a starfish and either cut or rip it in half, the two halves will regrow a entire starfish, complete with any physical abnormalities the orginal starfish possessed. Identical in DNA too.
Okay but suppose fully sentient beings could regrow themselves like starfish, it might still be clear that one of the two "regrown" beings was actually a new duplicate being with a new consciousness while the other is a continuation of the "original" consciousness.
A person is sliced in half "horizontally" around the waist into two parts. The upper half person continues to function mentally and continues to maintain his consciousness. Meanwhile the lower half part, since not yet having a head/brain, is now unconscious. But it regrows the entire upper half including his head and brain. Let's say this head and brain regrew to contain all the memories of the original pre-sliced person. Well in this example clearly the upper half regrown person is the original while the lower half person is a duplicate.
Of course if you sliced the person "vertically" in half from the head downward, it becomes somewhat more difficult to determine which half is the original. This is even more so if cutting in half vertically makes BOTH halves lose their consciousness and both halves then regain consciousness at a later time once fully regenerated into whole persons. In THIS case, it is certainly a possibility then that the ORIGINAL consciousness has died and two NEW consciousness but inheriting the memories of the original have been created!