The only thing they share is the blueprint by which their physical bodies were built. This is important. There seems to be a lot of confusion on this point. Cloning does not reproduce one's self, but only an identical twin of one's body. If you took one of your cells and cloned it, the result wouldn't be another one of you, but rather a twin brother or sister.
Identical twins have separate and distinct bodies.
They also have distinct souls. All creatures that normally possess souls would still have souls even when cloned. Sensations, thoughts, purposings, beliefs, and desires are all functions of a soul. A moment's reflection will show that material objects are not the kinds of things that possess features like thoughts or intentions. My table has no desires.